Case study background and problem formulations
Instructions for optimization with PSG MATLAB Subroutines.
CYCLE (PROBLEM 1 + PROBLEM 2):
Minimize Kantorovich-Rubinstein distance
subject to
Linearmulti = vector_of_fixed_probabilities (constraint on linking atoms between distributions)
Box constraints (lower bounds on probabilities)
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Linearmulti = set of linear functions specified by a matrix of scenarios
Box constraints = constraints on individual decision variables
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Minimize Kantorovich-Rubinstein distance
subject to
Linearmulti = vector_of_fixed_probabilities (constraint on linking atoms between distributions)
Box constraints (lower bounds on probabilities)
——————————————————————–
Linearmulti = set of linear functions specified by a matrix of scenarios
Box constraints = constraints on individual decision variables
——————————————————————–
| # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.14GHz (sec) | ||||
| Dataset | 30 | N/A | 0.1406680967 | 14.3 | |||
|---|---|---|---|---|---|---|---|
| Environments | |||||||
| Matlab Subroutines | Matlab Code | Data | |||||
PROBLEM 1: Minimize Sum of Euclidean Distances
Minimize Sum of Sqrt_Quadratic
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Sqrt_Quadratic = square root of quadratic function
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Minimize Sum of Sqrt_Quadratic
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Sqrt_Quadratic = square root of quadratic function
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| # of Variables | # of Scenarios | Objective Value | Solving Time, PC 3.14GHz (sec) | ||||
| Dataset | 20 | N/A | 0.1406681 | 0.03 | |||
|---|---|---|---|---|---|---|---|
| Environments | |||||||
| Run-File | Problem Statement | Data | Solution | ||||
CASE STUDY SUMMARY
Numerical algorithm in MATLAB environment for approximation of a discrete distribution in 2-dimensional space by some other discrete distribution with a smaller number of atoms. The approximation is done my minimizing the Kantorovich-Rubinstein distance between distributions. Positions and probabilities of atoms of the approximating distribution are variables of the optimization problem. The algorithm solves a sequence of optimization problems reducing the distance between distributions. Optimization problems are solved by calling PSG solver in MATLAB environment. One instance of Problem 1 (changing the positions of atoms) is presented in PSG Run-file environment. Problem statement and algorithm are described in Kuzmenko and Uryasev [1].
Numerical algorithm in MATLAB environment for approximation of a discrete distribution in 2-dimensional space by some other discrete distribution with a smaller number of atoms. The approximation is done my minimizing the Kantorovich-Rubinstein distance between distributions. Positions and probabilities of atoms of the approximating distribution are variables of the optimization problem. The algorithm solves a sequence of optimization problems reducing the distance between distributions. Optimization problems are solved by calling PSG solver in MATLAB environment. One instance of Problem 1 (changing the positions of atoms) is presented in PSG Run-file environment. Problem statement and algorithm are described in Kuzmenko and Uryasev [1].
References
[1] V. Kuzmenko and S. Uryasev. KANTOROVICH-RUBINSTEIN DISTANCE MINIMIZATION: APPLICATION TO LOCATION PROBLEMS. In Large Scale Optimization Applied to Supply Chain & Smart Manufacturing: Theory & Real Applications. Springer Optimization and Its Applications. 2019
[1] V. Kuzmenko and S. Uryasev. KANTOROVICH-RUBINSTEIN DISTANCE MINIMIZATION: APPLICATION TO LOCATION PROBLEMS. In Large Scale Optimization Applied to Supply Chain & Smart Manufacturing: Theory & Real Applications. Springer Optimization and Its Applications. 2019