** Case study background and problem formulations**

Instructions for optimization with PSG MATLAB Subroutines.

CYCLE (PROBLEM 1 + PROBLEM 2):

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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# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 30 | N/A | 0.1406680967 | 14.3 | |||
---|---|---|---|---|---|---|---|

Environments |
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Matlab Subroutines | Matlab Code | Data |

PROBLEM 1: Minimize Sum of

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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# of Variables |
# of Scenarios |
Objective Value |
Solving Time, PC 3.14GHz (sec) |
||||

Dataset | 20 | N/A | 0.1406681 | 0.03 | |||
---|---|---|---|---|---|---|---|

Environments |
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Run-File | Problem Statement | Data | Solution | ||||

R | R Code | Data |

**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].

References

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