MKL.cpp

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/*
 * This program is free software; you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation; either version 3 of the License, or
 * (at your option) any later version.
 *
 * Written (W) 2004-2009 Soeren Sonnenburg, Gunnar Raetsch
 *                       Alexander Zien, Marius Kloft, Chen Guohua
 * Copyright (C) 2009 Fraunhofer Institute FIRST and Max-Planck-Society
 * Copyright (C) 2010 Ryota Tomioka (University of Tokyo)
 */

#include <list>
#include <shogun/lib/Signal.h>
#include <shogun/classifier/mkl/MKL.h>
#include <shogun/classifier/svm/LibSVM.h>
#include <shogun/kernel/CombinedKernel.h>

#ifdef USE_GLPK
#include <glpk.h>
#endif

#ifdef USE_CPLEX
extern "C" {
#include <ilcplex/cplex.h>
}
#endif

using namespace shogun;

class CMKL::Self
{
public:
    void init() {
#ifdef USE_CPLEX
        lp_cplex = NULL ;
        env = NULL ;
#endif

#ifdef USE_GLPK
        lp_glpk = NULL;
        lp_glpk_parm = NULL;
#endif
    }

#ifdef USE_CPLEX

    bool init_cplex()
    {
        while (env==NULL)
        {
            SG_INFO("trying to initialize CPLEX\n")

            int status = 0; // calling external lib
            env = CPXopenCPLEX (&status);

            if ( env == NULL )
            {
                char  errmsg[1024];
                SG_WARNING("Could not open CPLEX environment.\n")
                CPXgeterrorstring (env, status, errmsg);
                SG_WARNING("%s", errmsg)
                SG_WARNING("retrying in 60 seconds\n")
                sleep(60);
            }
            else
            {
                // select dual simplex based optimization
                status = CPXsetintparam (env, CPX_PARAM_LPMETHOD, CPX_ALG_DUAL);
                if ( status )
                {
                SG_ERROR("Failure to select dual lp optimization, error %d.\n", status)
                }
                else
                {
                    status = CPXsetintparam (env, CPX_PARAM_DATACHECK, CPX_ON);
                    if ( status )
                    {
                        SG_ERROR("Failure to turn on data checking, error %d.\n", status)
                    }
                    else
                    {
                        lp_cplex = CPXcreateprob (env, &status, "light");

                        if ( lp_cplex == NULL )
                            SG_ERROR("Failed to create LP.\n")
                        else
                            CPXchgobjsen (env, lp_cplex, CPX_MIN);  /* Problem is minimization */
                    }
                }
            }
        }
        return (lp_cplex != NULL) && (env != NULL);
    }

    void set_qnorm_constraints(float64_t* beta, int32_t num_kernels)
    {
        ASSERT(num_kernels>0)

        float64_t* grad_beta=SG_MALLOC(float64_t, num_kernels);
        float64_t* hess_beta=SG_MALLOC(float64_t, num_kernels+1);
        float64_t* lin_term=SG_MALLOC(float64_t, num_kernels+1);
        int* ind=SG_MALLOC(int, num_kernels+1);

        //CMath::display_vector(beta, num_kernels, "beta");
        double const_term = 1-CMath::qsq(beta, num_kernels, mkl_norm);

        //SG_PRINT("const=%f\n", const_term)
        ASSERT(CMath::fequal(const_term, 0.0))

        for (int32_t i=0; i<num_kernels; i++)
        {
            grad_beta[i]=mkl_norm * pow(beta[i], mkl_norm-1);
            hess_beta[i]=0.5*mkl_norm*(mkl_norm-1) * pow(beta[i], mkl_norm-2);
            lin_term[i]=grad_beta[i] - 2*beta[i]*hess_beta[i];
            const_term+=grad_beta[i]*beta[i] - CMath::sq(beta[i])*hess_beta[i];
            ind[i]=i;
        }
        ind[num_kernels]=2*num_kernels;
        hess_beta[num_kernels]=0;
        lin_term[num_kernels]=0;

        int status=0;
        int num=CPXgetnumqconstrs (env, lp_cplex);

        if (num>0)
        {
            status = CPXdelqconstrs (env, lp_cplex, 0, 0);
            ASSERT(!status)
        }

        status = CPXaddqconstr (env, lp_cplex, num_kernels+1, num_kernels+1, const_term, 'L', ind, lin_term,
                ind, ind, hess_beta, NULL);
        ASSERT(!status)

        //CPXwriteprob (env, lp_cplex, "prob.lp", NULL);
        //CPXqpwrite (env, lp_cplex, "prob.qp");

        SG_FREE(grad_beta);
        SG_FREE(hess_beta);
        SG_FREE(lin_term);
        SG_FREE(ind);
    }

    bool cleanup_cplex(bool& lp_init)
    {
        bool result=false;

        if (lp_cplex)
        {
            int32_t status = CPXfreeprob(env, &lp_cplex);
            lp_cplex = NULL;
            lp_init = false;

            if (status)
                SG_WARNING("CPXfreeprob failed, error code %d.\n", status)
            else
                result = true;
        }

        if (env)
        {
            int32_t status = CPXcloseCPLEX (&env);
            env=NULL;

            if (status)
            {
                char  errmsg[1024];
                SG_WARNING("Could not close CPLEX environment.\n")
                CPXgeterrorstring (env, status, errmsg);
                SG_WARNING("%s", errmsg)
            }
            else
                result = true;
        }
        return result;
    }

    CPXENVptr     env;
    CPXLPptr      lp_cplex;
#endif

#ifdef USE_GLPK
    bool init_glpk()
    {
        lp_glpk = glp_create_prob();
        glp_set_obj_dir(lp_glpk, GLP_MIN);

        lp_glpk_parm = SG_MALLOC(glp_smcp, 1);
        glp_init_smcp(lp_glpk_parm);
        lp_glpk_parm->meth = GLP_DUAL;
        lp_glpk_parm->presolve = GLP_ON;

        glp_term_out(GLP_OFF);
        return (lp_glpk != NULL);
    }

    bool cleanup_glpk(bool& lp_init)
    {
        lp_init = false;
        if (lp_glpk)
            glp_delete_prob(lp_glpk);
        lp_glpk = NULL;
        SG_FREE(lp_glpk_parm);
        return true;
    }

    bool check_glp_status()
    {
        int status = glp_get_status(lp_glpk);

        if (status==GLP_INFEAS)
        {
            SG_SPRINT("solution is infeasible!\n")
            return false;
        }
        else if(status==GLP_NOFEAS)
        {
            SG_SPRINT("problem has no feasible solution!\n")
            return false;
        }
        return true;
    }

    glp_prob* lp_glpk;

    glp_smcp* lp_glpk_parm;
#endif
};

CMKL::CMKL(CSVM* s) : CSVM(), svm(NULL), C_mkl(0), mkl_norm(1), ent_lambda(0),
        mkl_block_norm(1),beta_local(NULL), mkl_iterations(0), mkl_epsilon(1e-5),
        interleaved_optimization(true), w_gap(1.0), rho(0), self()
{
    set_constraint_generator(s);
    self->init();

    SG_DEBUG("creating MKL object %p\n", this)
    lp_initialized = false ;
}

CMKL::~CMKL()
{
    // -- Delete beta_local for ElasticnetMKL
    SG_FREE(beta_local);

    SG_DEBUG("deleting MKL object %p\n", this)
    if (svm)
        svm->set_callback_function(NULL, NULL);
    SG_UNREF(svm);
}

void CMKL::init_solver()
{
#ifdef USE_CPLEX
    self->cleanup_cplex(lp_initialized);

    if (get_solver_type()==ST_CPLEX)
        self->init_cplex();
#endif

#ifdef USE_GLPK
    self->cleanup_glpk(lp_initialized);

    if (get_solver_type() == ST_GLPK)
        self->init_glpk();
#endif
}

bool CMKL::train_machine(CFeatures* data)
{
    ASSERT(kernel)
    ASSERT(m_labels && m_labels->get_num_labels())

    if (data)
    {
        if (m_labels->get_num_labels() != data->get_num_vectors())
        {
            SG_ERROR("%s::train_machine(): Number of training vectors (%d) does"
                    " not match number of labels (%d)\n", get_name(),
                    data->get_num_vectors(), m_labels->get_num_labels());
        }
        kernel->init(data, data);
    }

    init_training();
    if (!svm)
        SG_ERROR("No constraint generator (SVM) set\n")

    int32_t num_label=0;
    if (m_labels)
        num_label = m_labels->get_num_labels();

    SG_INFO("%d trainlabels (%ld)\n", num_label, m_labels)
    if (mkl_epsilon<=0)
        mkl_epsilon=1e-2 ;

    SG_INFO("mkl_epsilon = %1.1e\n", mkl_epsilon)
    SG_INFO("C_mkl = %1.1e\n", C_mkl)
    SG_INFO("mkl_norm = %1.3e\n", mkl_norm)
    SG_INFO("solver = %d\n", get_solver_type())
    SG_INFO("ent_lambda = %f\n", ent_lambda)
    SG_INFO("mkl_block_norm = %f\n", mkl_block_norm)

    int32_t num_weights = -1;
    int32_t num_kernels = kernel->get_num_subkernels();
    SG_INFO("num_kernels = %d\n", num_kernels)
    const float64_t* beta_const   = kernel->get_subkernel_weights(num_weights);
    float64_t* beta = SGVector<float64_t>::clone_vector(beta_const, num_weights);
    ASSERT(num_weights==num_kernels)

    if (get_solver_type()==ST_BLOCK_NORM &&
            mkl_block_norm>=1 &&
            mkl_block_norm<=2)
    {
        mkl_norm=mkl_block_norm/(2-mkl_block_norm);
        SG_WARNING("Switching to ST_DIRECT method with mkl_norm=%g\n", mkl_norm)
        set_solver_type(ST_DIRECT);
    }

    if (get_solver_type()==ST_ELASTICNET)
    {
      // -- Initialize subkernel weights for Elasticnet MKL
      SGVector<float64_t>::scale_vector(1/SGVector<float64_t>::qnorm(beta, num_kernels, 1.0), beta, num_kernels);

      SG_FREE(beta_local);
      beta_local = SGVector<float64_t>::clone_vector(beta, num_kernels);

      elasticnet_transform(beta, ent_lambda, num_kernels);
    }
    else
    {
        SGVector<float64_t>::scale_vector(1/SGVector<float64_t>::qnorm(beta, num_kernels, mkl_norm),
                beta, num_kernels); //q-norm = 1
    }

    kernel->set_subkernel_weights(SGVector<float64_t>(beta, num_kernels, false));
    SG_FREE(beta);

    svm->set_bias_enabled(get_bias_enabled());
    svm->set_epsilon(get_epsilon());
    svm->set_max_train_time(get_max_train_time());
    svm->set_nu(get_nu());
    svm->set_C(get_C1(), get_C2());
    svm->set_qpsize(get_qpsize());
    svm->set_shrinking_enabled(get_shrinking_enabled());
    svm->set_linadd_enabled(get_linadd_enabled());
    svm->set_batch_computation_enabled(get_batch_computation_enabled());
    svm->set_labels(m_labels);
    svm->set_kernel(kernel);

#ifdef USE_CPLEX
    self->cleanup_cplex();

    if (get_solver_type()==ST_CPLEX)
        self->init_cplex();
#endif

#ifdef USE_GLPK
    if (get_solver_type()==ST_GLPK)
        self->init_glpk();
#endif

    mkl_iterations = 0;
    CSignal::clear_cancel();

    training_time_clock.start();

    if (interleaved_optimization)
    {
        if (svm->get_classifier_type() != CT_LIGHT && svm->get_classifier_type() != CT_SVRLIGHT)
        {
            SG_ERROR("Interleaved MKL optimization is currently "
                    "only supported with SVMlight\n");
        }

        //Note that there is currently no safe way to properly resolve this
        //situation: the mkl object hands itself as a reference to the svm which
        //in turn increases the reference count of mkl and when done decreases
        //the count. Thus we have to protect the mkl object from deletion by
        //ref()'ing it before we set the callback function and should also
        //unref() it afterwards. However, when the reference count was zero
        //before this unref() might actually try to destroy this (crash ahead)
        //but if we don't actually unref() the object we might leak memory...
        //So as a workaround we only unref when the reference count was >1
        //before.
#ifdef USE_REFERENCE_COUNTING
        int32_t refs=this->ref();
#endif
        svm->set_callback_function(this, perform_mkl_step_helper);
        svm->train();
        SG_DONE()
        svm->set_callback_function(NULL, NULL);
#ifdef USE_REFERENCE_COUNTING
        if (refs>1)
            this->unref();
#endif
    }
    else
    {
        float64_t* sumw = SG_MALLOC(float64_t, num_kernels);



        while (true)
        {
            svm->train();

            float64_t suma=compute_sum_alpha();
            compute_sum_beta(sumw);

            if((training_time_clock.cur_time_diff()>get_max_train_time ())&&(get_max_train_time ()>0))
            {
                SG_SWARNING("MKL Algorithm terminates PREMATURELY due to current training time exceeding get_max_train_time ()= %f . It may have not converged yet!\n",get_max_train_time ())
                break;
            }


            mkl_iterations++;
            if (perform_mkl_step(sumw, suma) || CSignal::cancel_computations())
                break;
        }

        SG_FREE(sumw);
    }
#ifdef USE_CPLEX
    self->cleanup_cplex(lp_initialized);
#endif
#ifdef USE_GLPK
    self->cleanup_glpk(lp_initialized);
#endif

    int32_t nsv=svm->get_num_support_vectors();
    create_new_model(nsv);

    set_bias(svm->get_bias());
    for (int32_t i=0; i<nsv; i++)
    {
        set_alpha(i, svm->get_alpha(i));
        set_support_vector(i, svm->get_support_vector(i));
    }
    return true;
}


void CMKL::set_mkl_norm(float64_t norm)
{

  if (norm<1)
    SG_ERROR("Norm must be >= 1, e.g., 1-norm is the standard MKL; norms>1 nonsparse MKL\n")

  mkl_norm = norm;
}

void CMKL::set_elasticnet_lambda(float64_t lambda)
{
  if (lambda>1 || lambda<0)
    SG_ERROR("0<=lambda<=1\n")

  if (lambda==0)
    lambda = 1e-6;
  else if (lambda==1.0)
    lambda = 1.0-1e-6;

  ent_lambda=lambda;
}

void CMKL::set_mkl_block_norm(float64_t q)
{
  if (q<1)
    SG_ERROR("1<=q<=inf\n")

  mkl_block_norm=q;
}

bool CMKL::perform_mkl_step(
        const float64_t* sumw, float64_t suma)
{
    if((training_time_clock.cur_time_diff()>get_max_train_time ())&&(get_max_train_time ()>0))
    {
        SG_SWARNING("MKL Algorithm terminates PREMATURELY due to current training time exceeding get_max_train_time ()= %f . It may have not converged yet!\n",get_max_train_time ())
        return true;
    }

    int32_t num_kernels = kernel->get_num_subkernels();
    int32_t nweights=0;
    const float64_t* old_beta = kernel->get_subkernel_weights(nweights);
    ASSERT(nweights==num_kernels)
    float64_t* beta = SG_MALLOC(float64_t, num_kernels);

#if defined(USE_CPLEX) || defined(USE_GLPK)
    int32_t inner_iters=0;
#endif
    float64_t mkl_objective=0;

    mkl_objective=-suma;
    for (int32_t i=0; i<num_kernels; i++)
    {
        beta[i]=old_beta[i];
        mkl_objective+=old_beta[i]*sumw[i];
    }

    w_gap = CMath::abs(1-rho/mkl_objective) ;

    if( (w_gap >= mkl_epsilon) ||
        (get_solver_type()==ST_AUTO || get_solver_type()==ST_NEWTON || get_solver_type()==ST_DIRECT ) || get_solver_type()==ST_ELASTICNET || get_solver_type()==ST_BLOCK_NORM)
    {
        if (get_solver_type()==ST_AUTO || get_solver_type()==ST_DIRECT)
        {
            rho=compute_optimal_betas_directly(beta, old_beta, num_kernels, sumw, suma, mkl_objective);
        }
        else if (get_solver_type()==ST_BLOCK_NORM)
        {
            rho=compute_optimal_betas_block_norm(beta, old_beta, num_kernels, sumw, suma, mkl_objective);
        }
        else if (get_solver_type()==ST_ELASTICNET)
        {
            // -- Direct update of subkernel weights for ElasticnetMKL
            // Note that ElasticnetMKL solves a different optimization
            // problem from the rest of the solver types
            rho=compute_optimal_betas_elasticnet(beta, old_beta, num_kernels, sumw, suma, mkl_objective);
        }
        else if (get_solver_type()==ST_NEWTON)
            rho=compute_optimal_betas_newton(beta, old_beta, num_kernels, sumw, suma, mkl_objective);
#ifdef USE_CPLEX
        else if (get_solver_type()==ST_CPLEX)
            rho=compute_optimal_betas_via_cplex(beta, old_beta, num_kernels, sumw, suma, inner_iters);
#endif
#ifdef USE_GLPK
        else if (get_solver_type()==ST_GLPK)
            rho=compute_optimal_betas_via_glpk(beta, old_beta, num_kernels, sumw, suma, inner_iters);
#endif
        else
            SG_ERROR("Solver type not supported (not compiled in?)\n")

        w_gap = CMath::abs(1-rho/mkl_objective) ;
    }

    kernel->set_subkernel_weights(SGVector<float64_t>(beta, num_kernels, false));
    SG_FREE(beta);

    return converged();
}

float64_t CMKL::compute_optimal_betas_elasticnet(
  float64_t* beta, const float64_t* old_beta, const int32_t num_kernels,
  const float64_t* sumw, const float64_t suma,
  const float64_t mkl_objective )
{
    const float64_t epsRegul = 0.01;  // fraction of root mean squared deviation
    float64_t obj;
    float64_t Z;
    float64_t preR;
    int32_t p;
    int32_t nofKernelsGood;

    // --- optimal beta
    nofKernelsGood = num_kernels;

    Z = 0;
    for (p=0; p<num_kernels; ++p )
    {
        if (sumw[p] >= 0.0 && old_beta[p] >= 0.0 )
        {
            beta[p] = CMath::sqrt(sumw[p]*old_beta[p]*old_beta[p]);
            Z += beta[p];
        }
        else
        {
            beta[p] = 0.0;
            --nofKernelsGood;
        }
        ASSERT( beta[p] >= 0 )
    }

    // --- normalize
    SGVector<float64_t>::scale_vector(1.0/Z, beta, num_kernels);

    // --- regularize & renormalize

    preR = 0.0;
    for( p=0; p<num_kernels; ++p )
        preR += CMath::pow( beta_local[p] - beta[p], 2.0 );
    const float64_t R = CMath::sqrt( preR ) * epsRegul;
    if( !( R >= 0 ) )
    {
        SG_PRINT("MKL-direct: p = %.3f\n", 1.0 )
        SG_PRINT("MKL-direct: nofKernelsGood = %d\n", nofKernelsGood )
        SG_PRINT("MKL-direct: Z = %e\n", Z )
        SG_PRINT("MKL-direct: eps = %e\n", epsRegul )
        for( p=0; p<num_kernels; ++p )
        {
            const float64_t t = CMath::pow( beta_local[p] - beta[p], 2.0 );
            SG_PRINT("MKL-direct: t[%3d] = %e  ( diff = %e = %e - %e )\n", p, t, beta_local[p]-beta[p], beta_local[p], beta[p] )
        }
        SG_PRINT("MKL-direct: preR = %e\n", preR )
        SG_PRINT("MKL-direct: preR/p = %e\n", preR )
        SG_PRINT("MKL-direct: sqrt(preR/p) = %e\n", CMath::sqrt(preR) )
        SG_PRINT("MKL-direct: R = %e\n", R )
        SG_ERROR("Assertion R >= 0 failed!\n" )
    }

    Z = 0.0;
    for( p=0; p<num_kernels; ++p )
    {
        beta[p] += R;
        Z += beta[p];
        ASSERT( beta[p] >= 0 )
    }
    Z = CMath::pow( Z, -1.0 );
    ASSERT( Z >= 0 )
    for( p=0; p<num_kernels; ++p )
    {
        beta[p] *= Z;
        ASSERT( beta[p] >= 0.0 )
        if (beta[p] > 1.0 )
            beta[p] = 1.0;
    }

    // --- copy beta into beta_local
    for( p=0; p<num_kernels; ++p )
        beta_local[p] = beta[p];

    // --- elastic-net transform
    elasticnet_transform(beta, ent_lambda, num_kernels);

    // --- objective
    obj = -suma;
    for (p=0; p<num_kernels; ++p )
    {
        //obj += sumw[p] * old_beta[p]*old_beta[p] / beta[p];
        obj += sumw[p] * beta[p];
    }
    return obj;
}

void CMKL::elasticnet_dual(float64_t *ff, float64_t *gg, float64_t *hh,
        const float64_t &del, const float64_t* nm, int32_t len,
        const float64_t &lambda)
{
    std::list<int32_t> I;
    float64_t gam = 1.0-lambda;
    for (int32_t i=0; i<len;i++)
    {
        if (nm[i]>=CMath::sqrt(2*del*gam))
            I.push_back(i);
    }
    int32_t M=I.size();

    *ff=del;
    *gg=-(M*gam+lambda);
    *hh=0;
    for (std::list<int32_t>::iterator it=I.begin(); it!=I.end(); it++)
    {
        float64_t nmit = nm[*it];

        *ff += del*gam*CMath::pow(nmit/CMath::sqrt(2*del*gam)-1,2)/lambda;
        *gg += CMath::sqrt(gam/(2*del))*nmit;
        *hh += -0.5*CMath::sqrt(gam/(2*CMath::pow(del,3)))*nmit;
    }
}

// assumes that all constraints are satisfied
float64_t CMKL::compute_elasticnet_dual_objective()
{
    int32_t n=get_num_support_vectors();
    int32_t num_kernels = kernel->get_num_subkernels();
    float64_t mkl_obj=0;

    if (m_labels && kernel && kernel->get_kernel_type() == K_COMBINED)
    {
        // Compute Elastic-net dual
        float64_t* nm = SG_MALLOC(float64_t, num_kernels);
        float64_t del=0;


        int32_t k=0;
        for (index_t k_idx=0; k_idx<((CCombinedKernel*) kernel)->get_num_kernels(); k_idx++)
        {
            CKernel* kn = ((CCombinedKernel*) kernel)->get_kernel(k_idx);
            float64_t sum=0;
            for (int32_t i=0; i<n; i++)
            {
                int32_t ii=get_support_vector(i);

                for (int32_t j=0; j<n; j++)
                {
                    int32_t jj=get_support_vector(j);
                    sum+=get_alpha(i)*get_alpha(j)*kn->kernel(ii,jj);
                }
            }
            nm[k]= CMath::pow(sum, 0.5);
            del = CMath::max(del, nm[k]);

            // SG_PRINT("nm[%d]=%f\n",k,nm[k])
            k++;


            SG_UNREF(kn);
        }
        // initial delta
        del = del/CMath::sqrt(2*(1-ent_lambda));

        // Newton's method to optimize delta
        k=0;
        float64_t ff, gg, hh;
        elasticnet_dual(&ff, &gg, &hh, del, nm, num_kernels, ent_lambda);
        while (CMath::abs(gg)>+1e-8 && k<100)
        {
            float64_t ff_old = ff;
            float64_t gg_old = gg;
            float64_t del_old = del;

            // SG_PRINT("[%d] fval=%f gg=%f del=%f\n", k, ff, gg, del)

            float64_t step=1.0;
            do
            {
                del=del_old*CMath::exp(-step*gg/(hh*del+gg));
                elasticnet_dual(&ff, &gg, &hh, del, nm, num_kernels, ent_lambda);
                step/=2;
            } while(ff>ff_old+1e-4*gg_old*(del-del_old));

            k++;
        }
        mkl_obj=-ff;

        SG_FREE(nm);

        mkl_obj+=compute_sum_alpha();

    }
    else
        SG_ERROR("cannot compute objective, labels or kernel not set\n")

    return -mkl_obj;
}

float64_t CMKL::compute_optimal_betas_block_norm(
  float64_t* beta, const float64_t* old_beta, const int32_t num_kernels,
  const float64_t* sumw, const float64_t suma,
  const float64_t mkl_objective )
{
    float64_t obj;
    float64_t Z=0;
    int32_t p;

    // --- optimal beta
    for( p=0; p<num_kernels; ++p )
    {
        ASSERT(sumw[p]>=0)

        beta[p] = CMath::pow( sumw[p], -(2.0-mkl_block_norm)/(2.0-2.0*mkl_block_norm));
        Z+= CMath::pow( sumw[p], -(mkl_block_norm)/(2.0-2.0*mkl_block_norm));

        ASSERT( beta[p] >= 0 )
    }

    ASSERT(Z>=0)

    Z=1.0/CMath::pow(Z, (2.0-mkl_block_norm)/mkl_block_norm);

    for( p=0; p<num_kernels; ++p )
        beta[p] *= Z;

    // --- objective
    obj = -suma;
    for( p=0; p<num_kernels; ++p )
        obj += sumw[p] * beta[p];

    return obj;
}


float64_t CMKL::compute_optimal_betas_directly(
  float64_t* beta, const float64_t* old_beta, const int32_t num_kernels,
  const float64_t* sumw, const float64_t suma,
  const float64_t mkl_objective )
{
    const float64_t epsRegul = 0.01;  // fraction of root mean squared deviation
    float64_t obj;
    float64_t Z;
    float64_t preR;
    int32_t p;
    int32_t nofKernelsGood;

    // --- optimal beta
    nofKernelsGood = num_kernels;
    for( p=0; p<num_kernels; ++p )
    {
        //SG_PRINT("MKL-direct:  sumw[%3d] = %e  ( oldbeta = %e )\n", p, sumw[p], old_beta[p] )
        if( sumw[p] >= 0.0 && old_beta[p] >= 0.0 )
        {
            beta[p] = sumw[p] * old_beta[p]*old_beta[p] / mkl_norm;
            beta[p] = CMath::pow( beta[p], 1.0 / (mkl_norm+1.0) );
        }
        else
        {
            beta[p] = 0.0;
            --nofKernelsGood;
        }
        ASSERT( beta[p] >= 0 )
    }

    // --- normalize
    Z = 0.0;
    for( p=0; p<num_kernels; ++p )
        Z += CMath::pow( beta[p], mkl_norm );

    Z = CMath::pow( Z, -1.0/mkl_norm );
    ASSERT( Z >= 0 )
    for( p=0; p<num_kernels; ++p )
        beta[p] *= Z;

    // --- regularize & renormalize
    preR = 0.0;
    for( p=0; p<num_kernels; ++p )
        preR += CMath::sq( old_beta[p] - beta[p]);

    const float64_t R = CMath::sqrt( preR / mkl_norm ) * epsRegul;
    if( !( R >= 0 ) )
    {
        SG_PRINT("MKL-direct: p = %.3f\n", mkl_norm )
        SG_PRINT("MKL-direct: nofKernelsGood = %d\n", nofKernelsGood )
        SG_PRINT("MKL-direct: Z = %e\n", Z )
        SG_PRINT("MKL-direct: eps = %e\n", epsRegul )
        for( p=0; p<num_kernels; ++p )
        {
            const float64_t t = CMath::pow( old_beta[p] - beta[p], 2.0 );
            SG_PRINT("MKL-direct: t[%3d] = %e  ( diff = %e = %e - %e )\n", p, t, old_beta[p]-beta[p], old_beta[p], beta[p] )
        }
        SG_PRINT("MKL-direct: preR = %e\n", preR )
        SG_PRINT("MKL-direct: preR/p = %e\n", preR/mkl_norm )
        SG_PRINT("MKL-direct: sqrt(preR/p) = %e\n", CMath::sqrt(preR/mkl_norm) )
        SG_PRINT("MKL-direct: R = %e\n", R )
        SG_ERROR("Assertion R >= 0 failed!\n" )
    }

    Z = 0.0;
    for( p=0; p<num_kernels; ++p )
    {
        beta[p] += R;
        Z += CMath::pow( beta[p], mkl_norm );
        ASSERT( beta[p] >= 0 )
    }
    Z = CMath::pow( Z, -1.0/mkl_norm );
    ASSERT( Z >= 0 )
    for( p=0; p<num_kernels; ++p )
    {
        beta[p] *= Z;
        ASSERT( beta[p] >= 0.0 )
        if( beta[p] > 1.0 )
            beta[p] = 1.0;
    }

    // --- objective
    obj = -suma;
    for( p=0; p<num_kernels; ++p )
        obj += sumw[p] * beta[p];

    return obj;
}

float64_t CMKL::compute_optimal_betas_newton(float64_t* beta,
        const float64_t* old_beta, int32_t num_kernels,
        const float64_t* sumw, float64_t suma,
         float64_t mkl_objective)
{
    SG_DEBUG("MKL via NEWTON\n")

    if (mkl_norm==1)
        SG_ERROR("MKL via NEWTON works only for norms>1\n")

    const double epsBeta = 1e-32;
    const double epsGamma = 1e-12;
    const double epsWsq = 1e-12;
    const double epsNewt = 0.0001;
    const double epsStep = 1e-9;
    const int nofNewtonSteps = 3;
    const double hessRidge = 1e-6;
    const int inLogSpace = 0;

    const float64_t r = mkl_norm / ( mkl_norm - 1.0 );
    float64_t* newtDir = SG_MALLOC(float64_t,  num_kernels );
    float64_t* newtBeta = SG_MALLOC(float64_t,  num_kernels );
    //float64_t newtStep;
    float64_t stepSize;
    float64_t Z;
    float64_t obj;
    float64_t gamma;
    int32_t p;
    int i;

    // --- check beta
    Z = 0.0;
    for( p=0; p<num_kernels; ++p )
    {
        beta[p] = old_beta[p];
        if( !( beta[p] >= epsBeta ) )
            beta[p] = epsBeta;

        ASSERT( 0.0 <= beta[p] && beta[p] <= 1.0 )
        Z += CMath::pow( beta[p], mkl_norm );
    }

    Z = CMath::pow( Z, -1.0/mkl_norm );
    if( !( fabs(Z-1.0) <= epsGamma ) )
    {
        SG_WARNING("old_beta not normalized (diff=%e);  forcing normalization.  ", Z-1.0 )
        for( p=0; p<num_kernels; ++p )
        {
            beta[p] *= Z;
            if( beta[p] > 1.0 )
                beta[p] = 1.0;
            ASSERT( 0.0 <= beta[p] && beta[p] <= 1.0 )
        }
    }

    // --- compute gamma
    gamma = 0.0;
    for ( p=0; p<num_kernels; ++p )
    {
        if ( !( sumw[p] >= 0 ) )
        {
            if( !( sumw[p] >= -epsWsq ) )
                SG_WARNING("sumw[%d] = %e;  treated as 0.  ", p, sumw[p] )
            // should better recompute sumw[] !!!
        }
        else
        {
            ASSERT( sumw[p] >= 0 )
            //gamma += CMath::pow( sumw[p] * beta[p]*beta[p], r );
            gamma += CMath::pow( sumw[p] * beta[p]*beta[p] / mkl_norm, r );
        }
    }
    gamma = CMath::pow( gamma, 1.0/r ) / mkl_norm;
    ASSERT( gamma > -1e-9 )
    if( !( gamma > epsGamma ) )
    {
        SG_WARNING("bad gamma: %e;  set to %e.  ", gamma, epsGamma )
        // should better recompute sumw[] !!!
        gamma = epsGamma;
    }
    ASSERT( gamma >= epsGamma )
    //gamma = -gamma;

    // --- compute objective
    obj = 0.0;
    for( p=0; p<num_kernels; ++p )
    {
        obj += beta[p] * sumw[p];
        //obj += gamma/mkl_norm * CMath::pow( beta[p], mkl_norm );
    }
    if( !( obj >= 0.0 ) )
        SG_WARNING("negative objective: %e.  ", obj )
    //SG_PRINT("OBJ = %e.  \n", obj )


    // === perform Newton steps
    for (i = 0; i < nofNewtonSteps; ++i )
    {
        // --- compute Newton direction (Hessian is diagonal)
        const float64_t gqq1 = mkl_norm * (mkl_norm-1.0) * gamma;
        // newtStep = 0.0;
        for( p=0; p<num_kernels; ++p )
        {
            ASSERT( 0.0 <= beta[p] && beta[p] <= 1.0 )
            //const float halfw2p = ( sumw[p] >= 0.0 ) ? sumw[p] : 0.0;
            //const float64_t t1 = halfw2p*beta[p] - mkl_norm*gamma*CMath::pow(beta[p],mkl_norm);
            //const float64_t t2 = 2.0*halfw2p + gqq1*CMath::pow(beta[p],mkl_norm-1.0);
            const float halfw2p = ( sumw[p] >= 0.0 ) ? (sumw[p]*old_beta[p]*old_beta[p]) : 0.0;
            const float64_t t0 = halfw2p*beta[p] - mkl_norm*gamma*CMath::pow(beta[p],mkl_norm+2.0);
            const float64_t t1 = ( t0 < 0 ) ? 0.0 : t0;
            const float64_t t2 = 2.0*halfw2p + gqq1*CMath::pow(beta[p],mkl_norm+1.0);
            if( inLogSpace )
                newtDir[p] = t1 / ( t1 + t2*beta[p] + hessRidge );
            else
                newtDir[p] = ( t1 == 0.0 ) ? 0.0 : ( t1 / t2 );
            // newtStep += newtDir[p] * grad[p];
            ASSERT( newtDir[p] == newtDir[p] )
            //SG_PRINT("newtDir[%d] = %6.3f = %e / %e \n", p, newtDir[p], t1, t2 )
        }
        //CMath::display_vector( newtDir, num_kernels, "newton direction  " );
        //SG_PRINT("Newton step size = %e\n", Z )

        // --- line search
        stepSize = 1.0;
        while( stepSize >= epsStep )
        {
            // --- perform Newton step
            Z = 0.0;
            while( Z == 0.0 )
            {
                for( p=0; p<num_kernels; ++p )
                {
                    if( inLogSpace )
                        newtBeta[p] = beta[p] * CMath::exp( + stepSize * newtDir[p] );
                    else
                        newtBeta[p] = beta[p] + stepSize * newtDir[p];
                    if( !( newtBeta[p] >= epsBeta ) )
                        newtBeta[p] = epsBeta;
                    Z += CMath::pow( newtBeta[p], mkl_norm );
                }
                ASSERT( 0.0 <= Z )
                Z = CMath::pow( Z, -1.0/mkl_norm );
                if( Z == 0.0 )
                    stepSize /= 2.0;
            }

            // --- normalize new beta (wrt p-norm)
            ASSERT( 0.0 < Z )
            ASSERT( Z < CMath::INFTY )
            for( p=0; p<num_kernels; ++p )
            {
                newtBeta[p] *= Z;
                if( newtBeta[p] > 1.0 )
                {
                    //SG_WARNING("beta[%d] = %e;  set to 1.  ", p, beta[p] )
                    newtBeta[p] = 1.0;
                }
                ASSERT( 0.0 <= newtBeta[p] && newtBeta[p] <= 1.0 )
            }

            // --- objective increased?
            float64_t newtObj;
            newtObj = 0.0;
            for( p=0; p<num_kernels; ++p )
                newtObj += sumw[p] * old_beta[p]*old_beta[p] / newtBeta[p];
            //SG_PRINT("step = %.8f:  obj = %e -> %e.  \n", stepSize, obj, newtObj )
            if ( newtObj < obj - epsNewt*stepSize*obj )
            {
                for( p=0; p<num_kernels; ++p )
                    beta[p] = newtBeta[p];
                obj = newtObj;
                break;
            }
            stepSize /= 2.0;

        }

        if( stepSize < epsStep )
            break;
    }
    SG_FREE(newtDir);
    SG_FREE(newtBeta);

    // === return new objective
    obj = -suma;
    for( p=0; p<num_kernels; ++p )
        obj += beta[p] * sumw[p];
    return obj;
}



float64_t CMKL::compute_optimal_betas_via_cplex(float64_t* new_beta, const float64_t* old_beta, int32_t num_kernels,
          const float64_t* sumw, float64_t suma, int32_t& inner_iters)
{
    SG_DEBUG("MKL via CPLEX\n")

#ifdef USE_CPLEX
    ASSERT(new_beta)
    ASSERT(old_beta)

    int32_t NUMCOLS = 2*num_kernels + 1;
    double* x=SG_MALLOC(double, NUMCOLS);

    if (!lp_initialized)
    {
        SG_INFO("creating LP\n")

        double   obj[NUMCOLS]; /* calling external lib */
        double   lb[NUMCOLS]; /* calling external lib */
        double   ub[NUMCOLS]; /* calling external lib */

        for (int32_t i=0; i<2*num_kernels; i++)
        {
            obj[i]=0 ;
            lb[i]=0 ;
            ub[i]=1 ;
        }

        for (int32_t i=num_kernels; i<2*num_kernels; i++)
            obj[i]= C_mkl;

        obj[2*num_kernels]=1 ;
        lb[2*num_kernels]=-CPX_INFBOUND ;
        ub[2*num_kernels]=CPX_INFBOUND ;

        int status = CPXnewcols (self->env, self->lp_cplex, NUMCOLS, obj, lb, ub, NULL, NULL);
        if ( status ) {
            char  errmsg[1024];
            CPXgeterrorstring (self->env, status, errmsg);
            SG_ERROR("%s", errmsg)
        }

        // add constraint sum(w)=1;
        SG_INFO("adding the first row\n")
        int initial_rmatbeg[1]; /* calling external lib */
        int initial_rmatind[num_kernels+1]; /* calling external lib */
        double initial_rmatval[num_kernels+1]; /* calling ext lib */
        double initial_rhs[1]; /* calling external lib */
        char initial_sense[1];

        // 1-norm MKL
        if (mkl_norm==1)
        {
            initial_rmatbeg[0] = 0;
            initial_rhs[0]=1 ;     // rhs=1 ;
            initial_sense[0]='E' ; // equality

            // sparse matrix
            for (int32_t i=0; i<num_kernels; i++)
            {
                initial_rmatind[i]=i ; //index of non-zero element
                initial_rmatval[i]=1 ; //value of ...
            }
            initial_rmatind[num_kernels]=2*num_kernels ; //number of non-zero elements
            initial_rmatval[num_kernels]=0 ;

            status = CPXaddrows (self->env, self->lp_cplex, 0, 1, num_kernels+1,
                    initial_rhs, initial_sense, initial_rmatbeg,
                    initial_rmatind, initial_rmatval, NULL, NULL);

        }
        else // 2 and q-norm MKL
        {
            initial_rmatbeg[0] = 0;
            initial_rhs[0]=0 ;     // rhs=1 ;
            initial_sense[0]='L' ; // <=  (inequality)

            initial_rmatind[0]=2*num_kernels ;
            initial_rmatval[0]=0 ;

            status = CPXaddrows (self->env, self->lp_cplex, 0, 1, 1,
                    initial_rhs, initial_sense, initial_rmatbeg,
                    initial_rmatind, initial_rmatval, NULL, NULL);


            if (mkl_norm==2)
            {
                for (int32_t i=0; i<num_kernels; i++)
                {
                    initial_rmatind[i]=i ;
                    initial_rmatval[i]=1 ;
                }
                initial_rmatind[num_kernels]=2*num_kernels ;
                initial_rmatval[num_kernels]=0 ;

                status = CPXaddqconstr (self->env, self->lp_cplex, 0, num_kernels+1, 1.0, 'L', NULL, NULL,
                        initial_rmatind, initial_rmatind, initial_rmatval, NULL);
            }
        }


        if ( status )
            SG_ERROR("Failed to add the first row.\n")

        lp_initialized = true ;

        if (C_mkl!=0.0)
        {
            for (int32_t q=0; q<num_kernels-1; q++)
            {
                // add constraint w[i]-w[i+1]<s[i];
                // add constraint w[i+1]-w[i]<s[i];
                int rmatbeg[1]; /* calling external lib */
                int rmatind[3]; /* calling external lib */
                double rmatval[3]; /* calling external lib */
                double rhs[1]; /* calling external lib */
                char sense[1];

                rmatbeg[0] = 0;
                rhs[0]=0 ;     // rhs=0 ;
                sense[0]='L' ; // <=
                rmatind[0]=q ;
                rmatval[0]=1 ;
                rmatind[1]=q+1 ;
                rmatval[1]=-1 ;
                rmatind[2]=num_kernels+q ;
                rmatval[2]=-1 ;
                status = CPXaddrows (self->env, self->lp_cplex, 0, 1, 3,
                        rhs, sense, rmatbeg,
                        rmatind, rmatval, NULL, NULL);
                if ( status )
                    SG_ERROR("Failed to add a smothness row (1).\n")

                rmatbeg[0] = 0;
                rhs[0]=0 ;     // rhs=0 ;
                sense[0]='L' ; // <=
                rmatind[0]=q ;
                rmatval[0]=-1 ;
                rmatind[1]=q+1 ;
                rmatval[1]=1 ;
                rmatind[2]=num_kernels+q ;
                rmatval[2]=-1 ;
                status = CPXaddrows (self->env, self->lp_cplex, 0, 1, 3,
                        rhs, sense, rmatbeg,
                        rmatind, rmatval, NULL, NULL);
                if ( status )
                    SG_ERROR("Failed to add a smothness row (2).\n")
            }
        }
    }

    { // add the new row
        //SG_INFO("add the new row\n")

        int rmatbeg[1];
        int rmatind[num_kernels+1];
        double rmatval[num_kernels+1];
        double rhs[1];
        char sense[1];

        rmatbeg[0] = 0;
        if (mkl_norm==1)
            rhs[0]=0 ;
        else
            rhs[0]=-suma ;

        sense[0]='L' ;

        for (int32_t i=0; i<num_kernels; i++)
        {
            rmatind[i]=i ;
            if (mkl_norm==1)
                rmatval[i]=-(sumw[i]-suma) ;
            else
                rmatval[i]=-sumw[i];
        }
        rmatind[num_kernels]=2*num_kernels ;
        rmatval[num_kernels]=-1 ;

        int32_t status = CPXaddrows (self->env, self->lp_cplex, 0, 1, num_kernels+1,
                rhs, sense, rmatbeg,
                rmatind, rmatval, NULL, NULL);
        if ( status )
            SG_ERROR("Failed to add the new row.\n")
    }

    inner_iters=0;
    int status;
    {

        if (mkl_norm==1) // optimize 1 norm MKL
            status = CPXlpopt (self->env, self->lp_cplex);
        else if (mkl_norm==2) // optimize 2-norm MKL
            status = CPXbaropt(self->env, self->lp_cplex);
        else // q-norm MKL
        {
            float64_t* beta=SG_MALLOC(float64_t, 2*num_kernels+1);
            float64_t objval_old=1e-8; //some value to cause the loop to not terminate yet
            for (int32_t i=0; i<num_kernels; i++)
                beta[i]=old_beta[i];
            for (int32_t i=num_kernels; i<2*num_kernels+1; i++)
                beta[i]=0;

            while (true)
            {
                //int rows=CPXgetnumrows(env, lp_cplex);
                //int cols=CPXgetnumcols(env, lp_cplex);
                //SG_PRINT("rows:%d, cols:%d (kernel:%d)\n", rows, cols, num_kernels)
                CMath::scale_vector(1/CMath::qnorm(beta, num_kernels, mkl_norm), beta, num_kernels);

                set_qnorm_constraints(beta, num_kernels);

                status = CPXbaropt(self->env, self->lp_cplex);
                if ( status )
                    SG_ERROR("Failed to optimize Problem.\n")

                int solstat=0;
                double objval=0;
                status=CPXsolution(self->env, self->lp_cplex, &solstat, &objval,
                        (double*) beta, NULL, NULL, NULL);

                if ( status )
                {
                    CMath::display_vector(beta, num_kernels, "beta");
                    SG_ERROR("Failed to obtain solution.\n")
                }

                CMath::scale_vector(1/CMath::qnorm(beta, num_kernels, mkl_norm), beta, num_kernels);

                //SG_PRINT("[%d] %f (%f)\n", inner_iters, objval, objval_old)
                if ((1-abs(objval/objval_old) < 0.1*mkl_epsilon)) // && (inner_iters>2))
                    break;

                objval_old=objval;

                inner_iters++;
            }
            SG_FREE(beta);
        }

        if ( status )
            SG_ERROR("Failed to optimize Problem.\n")

        // obtain solution
        int32_t cur_numrows=(int32_t) CPXgetnumrows(self->env, self->lp_cplex);
        int32_t cur_numcols=(int32_t) CPXgetnumcols(self->env, self->lp_cplex);
        int32_t num_rows=cur_numrows;
        ASSERT(cur_numcols<=2*num_kernels+1)

        float64_t* slack=SG_MALLOC(float64_t, cur_numrows);
        float64_t* pi=SG_MALLOC(float64_t, cur_numrows);

        /* calling external lib */
        int solstat=0;
        double objval=0;

        if (mkl_norm==1)
        {
            status=CPXsolution(self->env, self->lp_cplex, &solstat, &objval,
                    (double*) x, (double*) pi, (double*) slack, NULL);
        }
        else
        {
            status=CPXsolution(self->env, self->lp_cplex, &solstat, &objval,
                    (double*) x, NULL, (double*) slack, NULL);
        }

        int32_t solution_ok = (!status) ;
        if ( status )
            SG_ERROR("Failed to obtain solution.\n")

        int32_t num_active_rows=0 ;
        if (solution_ok)
        {
            /* 1 norm mkl */
            float64_t max_slack = -CMath::INFTY ;
            int32_t max_idx = -1 ;
            int32_t start_row = 1 ;
            if (C_mkl!=0.0)
                start_row+=2*(num_kernels-1);

            for (int32_t i = start_row; i < cur_numrows; i++)  // skip first
            {
                if (mkl_norm==1)
                {
                    if ((pi[i]!=0))
                        num_active_rows++ ;
                    else
                    {
                        if (slack[i]>max_slack)
                        {
                            max_slack=slack[i] ;
                            max_idx=i ;
                        }
                    }
                }
                else // 2-norm or general q-norm
                {
                    if ((CMath::abs(slack[i])<1e-6))
                        num_active_rows++ ;
                    else
                    {
                        if (slack[i]>max_slack)
                        {
                            max_slack=slack[i] ;
                            max_idx=i ;
                        }
                    }
                }
            }

            // have at most max(100,num_active_rows*2) rows, if not, remove one
            if ( (num_rows-start_row>CMath::max(100,2*num_active_rows)) && (max_idx!=-1))
            {
                //SG_INFO("-%i(%i,%i)",max_idx,start_row,num_rows)
                status = CPXdelrows (self->env, self->lp_cplex, max_idx, max_idx) ;
                if ( status )
                    SG_ERROR("Failed to remove an old row.\n")
            }

            //CMath::display_vector(x, num_kernels, "beta");

            rho = -x[2*num_kernels] ;
            SG_FREE(pi);
            SG_FREE(slack);

        }
        else
        {
            /* then something is wrong and we rather
            stop sooner than later */
            rho = 1 ;
        }
    }
    for (int32_t i=0; i<num_kernels; i++)
        new_beta[i]=x[i];

    SG_FREE(x);
#else
    SG_ERROR("Cplex not enabled at compile time\n")
#endif
    return rho;
}

float64_t CMKL::compute_optimal_betas_via_glpk(float64_t* beta, const float64_t* old_beta,
        int num_kernels, const float64_t* sumw, float64_t suma, int32_t& inner_iters)
{
    SG_DEBUG("MKL via GLPK\n")

    if (mkl_norm!=1)
        SG_ERROR("MKL via GLPK works only for norm=1\n")

    float64_t obj=1.0;
#ifdef USE_GLPK
    int32_t NUMCOLS = 2*num_kernels + 1 ;
    if (!lp_initialized)
    {
        SG_INFO("creating LP\n")

        //set obj function, note glpk indexing is 1-based
        glp_add_cols(self->lp_glpk, NUMCOLS);
        for (int i=1; i<=2*num_kernels; i++)
        {
            glp_set_obj_coef(self->lp_glpk, i, 0);
            glp_set_col_bnds(self->lp_glpk, i, GLP_DB, 0, 1);
        }
        for (int i=num_kernels+1; i<=2*num_kernels; i++)
        {
            glp_set_obj_coef(self->lp_glpk, i, C_mkl);
        }
        glp_set_obj_coef(self->lp_glpk, NUMCOLS, 1);
        glp_set_col_bnds(self->lp_glpk, NUMCOLS, GLP_FR, 0,0); //unbound

        //add first row. sum[w]=1
        int row_index = glp_add_rows(self->lp_glpk, 1);
        int* ind = SG_MALLOC(int, num_kernels+2);
        float64_t* val = SG_MALLOC(float64_t, num_kernels+2);
        for (int i=1; i<=num_kernels; i++)
        {
            ind[i] = i;
            val[i] = 1;
        }
        ind[num_kernels+1] = NUMCOLS;
        val[num_kernels+1] = 0;
        glp_set_mat_row(self->lp_glpk, row_index, num_kernels, ind, val);
        glp_set_row_bnds(self->lp_glpk, row_index, GLP_FX, 1, 1);
        SG_FREE(val);
        SG_FREE(ind);

        lp_initialized = true;

        if (C_mkl!=0.0)
        {
            for (int32_t q=1; q<num_kernels; q++)
            {
                int mat_ind[4];
                float64_t mat_val[4];
                int mat_row_index = glp_add_rows(self->lp_glpk, 2);
                mat_ind[1] = q;
                mat_val[1] = 1;
                mat_ind[2] = q+1;
                mat_val[2] = -1;
                mat_ind[3] = num_kernels+q;
                mat_val[3] = -1;
                glp_set_mat_row(self->lp_glpk, mat_row_index, 3, mat_ind, mat_val);
                glp_set_row_bnds(self->lp_glpk, mat_row_index, GLP_UP, 0, 0);
                mat_val[1] = -1;
                mat_val[2] = 1;
                glp_set_mat_row(self->lp_glpk, mat_row_index+1, 3, mat_ind, mat_val);
                glp_set_row_bnds(self->lp_glpk, mat_row_index+1, GLP_UP, 0, 0);
            }
        }
    }

    int* ind=SG_MALLOC(int,num_kernels+2);
    float64_t* val=SG_MALLOC(float64_t, num_kernels+2);
    int row_index = glp_add_rows(self->lp_glpk, 1);
    for (int32_t i=1; i<=num_kernels; i++)
    {
        ind[i] = i;
        val[i] = -(sumw[i-1]-suma);
    }
    ind[num_kernels+1] = 2*num_kernels+1;
    val[num_kernels+1] = -1;
    glp_set_mat_row(self->lp_glpk, row_index, num_kernels+1, ind, val);
    glp_set_row_bnds(self->lp_glpk, row_index, GLP_UP, 0, 0);
    SG_FREE(ind);
    SG_FREE(val);

    //optimize
    glp_simplex(self->lp_glpk, self->lp_glpk_parm);
    bool res = self->check_glp_status();
    if (!res)
        SG_ERROR("Failed to optimize Problem.\n")

    int32_t cur_numrows = glp_get_num_rows(self->lp_glpk);
    int32_t cur_numcols = glp_get_num_cols(self->lp_glpk);
    int32_t num_rows=cur_numrows;
    ASSERT(cur_numcols<=2*num_kernels+1)

    float64_t* col_primal = SG_MALLOC(float64_t, cur_numcols);
    float64_t* row_primal = SG_MALLOC(float64_t, cur_numrows);
    float64_t* row_dual = SG_MALLOC(float64_t, cur_numrows);

    for (int i=0; i<cur_numrows; i++)
    {
        row_primal[i] = glp_get_row_prim(self->lp_glpk, i+1);
        row_dual[i] = glp_get_row_dual(self->lp_glpk, i+1);
    }
    for (int i=0; i<cur_numcols; i++)
        col_primal[i] = glp_get_col_prim(self->lp_glpk, i+1);

    obj = -col_primal[2*num_kernels];

    for (int i=0; i<num_kernels; i++)
        beta[i] = col_primal[i];

    int32_t num_active_rows=0;
    if(res)
    {
        float64_t max_slack = CMath::INFTY;
        int32_t max_idx = -1;
        int32_t start_row = 1;
        if (C_mkl!=0.0)
            start_row += 2*(num_kernels-1);

        for (int32_t i= start_row; i<cur_numrows; i++)
        {
            if (row_dual[i]!=0)
                num_active_rows++;
            else
            {
                if (row_primal[i]<max_slack)
                {
                    max_slack = row_primal[i];
                    max_idx = i;
                }
            }
        }

        if ((num_rows-start_row>CMath::max(100, 2*num_active_rows)) && max_idx!=-1)
        {
            int del_rows[2];
            del_rows[1] = max_idx+1;
            glp_del_rows(self->lp_glpk, 1, del_rows);
        }
    }

    SG_FREE(row_dual);
    SG_FREE(row_primal);
    SG_FREE(col_primal);
#else
    SG_ERROR("Glpk not enabled at compile time\n")
#endif

    return obj;
}

void CMKL::compute_sum_beta(float64_t* sumw)
{
    ASSERT(sumw)
    ASSERT(svm)

    int32_t nsv=svm->get_num_support_vectors();
    int32_t num_kernels = kernel->get_num_subkernels();
    SGVector<float64_t> beta=SGVector<float64_t>(num_kernels);
    int32_t nweights=0;
    const float64_t* old_beta = kernel->get_subkernel_weights(nweights);
    ASSERT(nweights==num_kernels)
    ASSERT(old_beta)

    for (int32_t i=0; i<num_kernels; i++)
    {
        beta.vector[i]=0;
        sumw[i]=0;
    }

    for (int32_t n=0; n<num_kernels; n++)
    {
        beta.vector[n]=1.0;
        /* this only copies the value of the first entry of this array
         * so it may be freed safely afterwards. */
        kernel->set_subkernel_weights(beta);

        for (int32_t i=0; i<nsv; i++)
        {
            int32_t ii=svm->get_support_vector(i);

            for (int32_t j=0; j<nsv; j++)
            {
                int32_t jj=svm->get_support_vector(j);
                sumw[n]+=0.5*svm->get_alpha(i)*svm->get_alpha(j)*kernel->kernel(ii,jj);
            }
        }
        beta[n]=0.0;
    }

    mkl_iterations++;
    kernel->set_subkernel_weights(SGVector<float64_t>( (float64_t*) old_beta, num_kernels, false));
}


// assumes that all constraints are satisfied
float64_t CMKL::compute_mkl_dual_objective()
{
    if (get_solver_type()==ST_ELASTICNET)
    {
        // -- Compute ElasticnetMKL dual objective
        return compute_elasticnet_dual_objective();
    }

    int32_t n=get_num_support_vectors();
    float64_t mkl_obj=0;

    if (m_labels && kernel && kernel->get_kernel_type() == K_COMBINED)
    {
        for (index_t k_idx=0; k_idx<((CCombinedKernel*) kernel)->get_num_kernels(); k_idx++)
        {
            CKernel* kn = ((CCombinedKernel*) kernel)->get_kernel(k_idx);
            float64_t sum=0;
            for (int32_t i=0; i<n; i++)
            {
                int32_t ii=get_support_vector(i);

                for (int32_t j=0; j<n; j++)
                {
                    int32_t jj=get_support_vector(j);
                    sum+=get_alpha(i)*get_alpha(j)*kn->kernel(ii,jj);
                }
            }

            if (mkl_norm==1.0)
                mkl_obj = CMath::max(mkl_obj, sum);
            else
                mkl_obj += CMath::pow(sum, mkl_norm/(mkl_norm-1));

            SG_UNREF(kn);
        }

        if (mkl_norm==1.0)
            mkl_obj=-0.5*mkl_obj;
        else
            mkl_obj= -0.5*CMath::pow(mkl_obj, (mkl_norm-1)/mkl_norm);

        mkl_obj+=compute_sum_alpha();
    }
    else
        SG_ERROR("cannot compute objective, labels or kernel not set\n")

    return -mkl_obj;
}