Solution in MATLAB Environment
Contents
Figure 1. Problem in PSG Toolbox Window
Case study Portfolio Optimization with Second Orders Stochastic Dominance Constraints (see Formal Problem Statement) in MATLAB Environment is solved with tbpsg_run PSG function.
Main MATLAB code is in file CS_SSD_Portfolio_Optimization_Toolbox.m.
Data are saved in file SSD_Portfolio_Optimization_data_Toolbox.mat:
| • | Scenarios of rates of returns (matrix_sde) used in (CS.1), (CS.2) and matrix for budget constraint (matrix_budget) used in (CS.3) (in toolboxstruc_arr structure). |
| • | Benchmark portfolio rates of returns used in (CS.2) (in benchmark_sd variable). |
| • | Calculated upper bounds for constraints (CS.2) (in ubound_sd variable ). |
Let us describe the main operations. To run case study you need to do the following main steps:
In file CS_SSD_Portfolio_Optimization_Toolbox.m:
clc;
clear;
Load problem data:
load('.\SSD_Portfolio_Optimization_data_Toolbox.mat');
Save variables from structure toolboxstruc_arr to Workspace:
tbpsg_export_to_workspace(toolboxstruc_arr);
J=size(toolboxstruc_arr(1).data.data,1); % A number of scenarios
I=size(toolboxstruc_arr(1).data.data,2); % A number of variables
Find redundant constraints:
ic=[];
for j=2:J
for j1=1:j-1
if ubound_sd(j)>=ubound_sd(j1) && benchmark_sd(j)<=benchmark_sd(j1), ic=[ic,j]; break, end
if ubound_sd(j)<=ubound_sd(j1) && benchmark_sd(j)>=benchmark_sd(j1), ic=[ic,j1]; end
end
end
ubound_sd_loc=ubound_sd;
benchmark_sd_loc=benchmark_sd;
ubound_sd_loc(ic) = [];
benchmark_sd_loc(ic) = [];
Create problem statement:
problem_head=['maximize', char(10),...
' linear(matrix_sde)', char(10),...
'Constraint: = 1', char(10),...
' linear(matrix_budget)'];
problem_newc='';
problem_newc=[problem_newc ...
'MultiConstraint: <= vector_ub',char(10),...
' pm_pen(vector_th, matrix_sde)',char(10)];
problem_box=[ 'Box: >=0, <=1',char(10),...
'Solver: precision = 9'];
problem_statement=[problem_head,char(10), problem_newc, problem_box];
disp(sprintf('Number of non-redundand constraints %g',size(ic,2)));
clear problem_head problem_newc problem_box j j1 ic;
Pack vector with upper bounds in constraints to PSG Toolbox structure:
toolboxstruc_arr(3) = tbpsg_vector_pack('vector_ub', ubound_sd_loc);
Pack vector with parameters of pm_pen function to PSG Toolbox structure:
toolboxstruc_arr(4) = tbpsg_vector_pack('vector_th', -benchmark_sd_loc);
% Uncomment the following line to open the problem in Toolbox Window:
% tbpsg_toolbox(problem_statement,toolboxstruc_arr);
Optimize problem:
[solution_SSD, outargstruc_SSD] = tbpsg_run(problem_statement, toolboxstruc_arr);
Display main results:
disp(' '); disp(' ')
disp(' Report ')
disp(' ')
disp('Solution_SSD ')
dx = strfind(solution_SSD, 'Constraint:');
disp(solution_SSD(1:dx(1)-1));
clear dx;
Optimal_point = tbpsg_optimal_point_data(solution_SSD, outargstruc_SSD)';
disp('Optimal point:');
for i=1:I; disp(['x', int2str(i),' = ' num2str(Optimal_point(i),7)]);
end
clear i; disp(' ')
Problem: problem_ssd, solution_status = optimal
Timing: Data_loading_time = 0.51, Preprocessing_time = 0.02, Solving_time = 0.96
Variables: optimal_point = point_problem_ssd
Objective: objective = 1.865255596763E-02 Gap: = -2.740863092043E-16
x1 = 0
x2 = 0
x3 = 0
x4 = 0.02911967
x5 = 0
x6 = 0
x7 = 0
x8 = 0
x9 = 0
x10 = 0
x11 = 0
x12 = 0
x13 = 0
x14 = 0
x15 = 0
x16 = 0
x17 = 0
x18 = 0
x19 = 0
x20 = 0
x21 = 0.2616667
x22 = 0
x23 = 0
x24 = 0
x25 = 0
x26 = 0
x27 = 0
x28 = 0
x29 = 0
x30 = 0
x31 = 0
x32 = 0
x33 = 0
x34 = 0
x35 = 0
x36 = 0
x37 = 0
x38 = 0
x39 = 0
x40 = 0
x41 = 0
x42 = 0
x43 = 0
x44 = 0
x45 = 0.2527069
x46 = 0
x47 = 0
x48 = 0
x49 = 0
x50 = 0
x51 = 0
x52 = 0
x53 = 0
x54 = 0
x55 = 0
x56 = 0.3583057
x57 = 0
x58 = 0.07687519
x59 = 0
x60 = 0
x61 = 0
x62 = 0
x63 = 0
x64 = 0
x65 = 0
x66 = 0
x67 = 0
x68 = 0
x69 = 0
x70 = 0
x71 = 0
x72 = 0
x73 = 0
x74 = 0
x75 = 0
x76 = 0.02132584
Figure 1. Problem in PSG Toolbox Window
