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** Case study background and problem formulations**

**Problem 1a: L2-norm geometric margin minimization formulation of Var-SVM**

for {solver_exp} = {CAR}, {VAN}, {BULDOZER}, {TANK}

minimize quadratic

subject to

var_risk

linearmulti

solver: solver_exp, precision = 9, stages = 30

end for

——————————————————————–

quadratic = quadratic function specified by a unit matrix

var_risk = Value-at-Risk specified by a matrix of scenarios

linearmulti = set of linear functions specified by a matrix of scenarios

——————————————————————–

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value (BULDOZER) | Solving Time (BULDOZER), PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 2 | 100 | 18.702062913380 | 0.22 |

Dataset2 | Problem Statement | Data | Solution | 3 | 100 | 4.012808007875 | 0.25 |

**Problem 1b: L2-norm structural risk minimization formulation of Var-SVM**

for {solver_exp} = {CAR}, {VAN}, {BULDOZER}, {TANK}

minimize quadratic + var

subject to

linearmulti

solver: solver_exp, precision = 9, stages = 30

end for

——————————————————————–

quadratic = quadratic function specified by a unit matrix

var_risk = Value-at-Risk specified by a matrix of scenarios

linearmulti = set of linear functions specified by a matrix of scenarios

——————————————————————–

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value (BULDOZER) | Solving Time (BULDOZER), PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 2 | 100 | -0.013367514210 | 0.22 |

Dataset2 | Problem Statement | Data | Solution | 3 | 100 | -0.062300476773 | 0.25 |

**Problem 2a: L1-norm geometric margin minimization formulation of Var-SVM**

for {solver_exp} = {CAR}, {VAN}, {BULDOZER}, {TANK}

minimize polynom_abs

subject to

var_risk

linearmulti

solver: solver_exp, precision = 9, stages = 30

end for

——————————————————————–

polynom_abs = L1-norm function

var_risk = Value-at-Risk specified by a matrix of scenarios

linearmulti = set of linear functions specified by a matrix of scenarios

——————————————————————–

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value (BULDOZER) | Solving Time (BULDOZER), PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 2 | 100 | 6.115910927534 | 0.07 |

Dataset2 | Problem Statement | Data | Solution | 3 | 100 | 4.100144111911 | 0.24 |

**Problem 2b: L1-norm structural risk minimization formulation of Var-SVM**

for {solver_exp} = {CAR}, {VAN}, {BULDOZER}, {TANK}

minimize polynom_abs + var_risk

subject to

linearmulti

solver: solver_exp, precision = 9, stages = 30

end for

——————————————————————–

polynom_abs = L1-norm function

var_risk = Value-at-Risk specified by a matrix of scenarios

linearmulti = set of linear functions specified by a matrix of scenarios

——————————————————————–

Data and solution in Run-File Environment

Problem Datasets | # of Variables | # of Scenarios | Objective Value (BULDOZER) | Solving Time (BULDOZER), PC 2.66GHz (sec) | |||
---|---|---|---|---|---|---|---|

Dataset1 | Problem Statement | Data | Solution | 2 | 100 | -8.259128038970 | 0.28 |

Dataset2 | Problem Statement | Data | Solution | 3 | 100 | -26.654598309919 | 0.27 |

**CASE STUDY SUMMARY**

This case study compares performance of several software packages for finding a solution to the Value-at-Risk Support Vector Machine (Var-SVM) classification problem. Given a training dataset , consisting of Q points with feature vectors and class labels , an SVM classification problem is to build a hyperplain in the feature space, that maximizes the geometric margin between the classes and the hyperplain. The original SVM was introduced by Vapnik [1979] for the case of maximization of an L2-norm margin and further reduced to a convex quadratic programming problem (QP) as long as the data set was separable. Nowadays, this formulation is known as a hard-margin SVM. In the case of a non-separable data set Cortes and Vapnik [1995] introduced the so-called soft-margin SVM, which can be viewed as a structural risk minimization problem, i.e., as a trade-off between an empirical classication error term and a regularization term, which is added to control overtting. Tsyurmasto et al. [2013] considers a generalization of the maximum margin approach corresponding to a particular choice of an empirical risk functional and provides a set of examples for such generalizations including hard- and soft-margin SVMs and nu-SVM. They also show, that for a positive homogeneous empirical risk functional both maximum margin and minimum structural risk L2-norm formulations provide the same solution, i.e., the same separating hyperplane. Zdanovskaya et al. [2014] provide an extension of this result for the case of L1-norm. Tsyurmasto et al. [2014] proposed a Var-SVM which corresponds to a positive homogeneous Value-at-Risk (VaR) empirical risk function. The main advantage of the Var-SVM is that it is stable to data outliers. The main disadvantage of the Var-SVM is that the problem is nonconvex and therefore is hard to solve. In this case study we consider four different formulations of Var-SVMs: 1a) maximum margin in L1-norm formulation, 1b) minimum structural risk in L1-norm formulation, 2a) maximum margin in L2-norm formulation, 2b) minimum structural risk in L2-norm formulation. To find a heuristic solution of the above Var-SVMs we use Portfolio Safeguard (PSG), since it has a preprogrammed Value-at-Risk function (VaR). In order to find an optimal solution of the above Var-SVMs we employ a mixed-integer linear programming representation (MILP representation) of VaR empirical risk functional proposed in Zdanovskaya et al. [2014]. This MILP representation is known to be NP-hard (see Benati and Rizzi [2007]). We further use such commercial solvers as Gurobi and Xpress to solve the derived Var-SVMs problems via the Branch&Bound algorithm. Our computational experiments show, that it may take Gurobi and Xpress up to several hours to find an optimal solution of Var-SVMs even for small datasets. Whereas it takes less than a second for PSG to find a corresponding heuristic solution.

**References**