Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solution of Robust Linear Optimization Problems

Parthasarathi Mondal, Akshay Kumar Ojha

TL;DR

A mathematical model is developed on the solution strategy for robust linear optimization problems, where the constraints only are associated with uncertainties, where the constraints only are associated with uncertainties.

Abstract

Robust optimization(RO) is an important tool for handling optimization problem with uncertainty. The main objective of RO is to solve optimization problems due to uncertainty associated with constraints satisfying all realizations of uncertain values within a given uncertainty set. The challenge of RO is to reformulate the constraints so that the uncertain optimization problem is transformed into a tractable deterministic form. In this paper, we have given more emphasis to study the robust counterpart(RC) of the RO problems and have developed a mathematical model on the solution strategy for robust linear optimization problems, where the constraints only are associated with uncertainties. The box and ellipsoidal uncertainty sets are considered and some illustrative numerical examples have been solved in each corresponding case for validating our proposed method.

Solution of Robust Linear Optimization Problems

TL;DR

A mathematical model is developed on the solution strategy for robust linear optimization problems, where the constraints only are associated with uncertainties, where the constraints only are associated with uncertainties.

Abstract

Robust optimization(RO) is an important tool for handling optimization problem with uncertainty. The main objective of RO is to solve optimization problems due to uncertainty associated with constraints satisfying all realizations of uncertain values within a given uncertainty set. The challenge of RO is to reformulate the constraints so that the uncertain optimization problem is transformed into a tractable deterministic form. In this paper, we have given more emphasis to study the robust counterpart(RC) of the RO problems and have developed a mathematical model on the solution strategy for robust linear optimization problems, where the constraints only are associated with uncertainties. The box and ellipsoidal uncertainty sets are considered and some illustrative numerical examples have been solved in each corresponding case for validating our proposed method.

Paper Structure

This paper contains 21 sections, 1 theorem, 54 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminary Discussions of Basic Concepts
  3. Some Basic assumptions:
  4. Important results and definitions
  5. Robust counterpart(RC)
  6. Uncertain set and perturbation
  7. Ben-Tal and Nemirovski approach to robust optimization 9
  8. Proposed approach and methodology
  9. Methodologies for solving a robust linear optimization problem
  10. Robust reformulation approach
  11. Uncertain RO problem with single uncertainty-affected constraint
  12. Uncertain RO problem with two uncertainty-affected constraints
  13. Robust discretisation technique
  14. Interval uncertainty
  15. Result Analysis
  16. ...and 6 more sections

Key Result

Theorem 1

Let P be an RO optimization problem with constraint-wise uncertainty. Then the RC of P is feasible if and only if all the instances of P are feasible, and its robust optimal value is given by, ${\sup c^Tx}_{(A,b)\in\mathcal{U}}$

Figures (3)

  • Figure 1: Grid points of box region
  • Figure 2: Grid points of ellipsoidal region
  • Figure 3: Simultaneous grid points of box and ellipse

Theorems & Definitions (5)

  • Definition 1
  • Definition 2: Robust value
  • Definition 3
  • Theorem 1
  • proof