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Polynomial algorithm for the disjoint bilinear programming problem with an acute-angled polytope for a disjoint subset

Dmitrii Lozovanu

TL;DR

The paper tackles disjoint bilinear programming when one disjoint subset forms an acute-angled (perfect) polyhedron. It develops a rigorous optimality criterion and two polynomial-time algorithms (Algorithm 1 and Algorithm 2) that exploit the polyhedral structure to verify feasibility and compute optimal solutions. The results connect disjoint bilinear programming to min–max linear problems with interdependent variables and enable efficient solutions for boolean linear programming and piecewise linear concave programming under the perfect-subset assumption. The work highlights a path to tractable solutions for certain NP-hard-looking problems by structural restrictions and provides a framework that unifies theory and practical algorithms for these cases.

Abstract

We consider the disjoint bilinear programming problem in which one of the disjoint subsets has the structure of an acute-angled polytope. An optimality criterion for such a problem is formulated and proved, and based on this, a polynomial algorithm for its solving is proposed and grounded. We show that the proposed algorithm can be efficiently used for studying and solving the boolean linear programming problem and the piecewise linear concave programming problem.

Polynomial algorithm for the disjoint bilinear programming problem with an acute-angled polytope for a disjoint subset

TL;DR

The paper tackles disjoint bilinear programming when one disjoint subset forms an acute-angled (perfect) polyhedron. It develops a rigorous optimality criterion and two polynomial-time algorithms (Algorithm 1 and Algorithm 2) that exploit the polyhedral structure to verify feasibility and compute optimal solutions. The results connect disjoint bilinear programming to min–max linear problems with interdependent variables and enable efficient solutions for boolean linear programming and piecewise linear concave programming under the perfect-subset assumption. The work highlights a path to tractable solutions for certain NP-hard-looking problems by structural restrictions and provides a framework that unifies theory and practical algorithms for these cases.

Abstract

We consider the disjoint bilinear programming problem in which one of the disjoint subsets has the structure of an acute-angled polytope. An optimality criterion for such a problem is formulated and proved, and based on this, a polynomial algorithm for its solving is proposed and grounded. We show that the proposed algorithm can be efficiently used for studying and solving the boolean linear programming problem and the piecewise linear concave programming problem.

Paper Structure

This paper contains 20 sections, 36 theorems, 197 equations.

Table of Contents

  1. Introduction and problem formulation
  2. Disjoint bilinear programming with a perfect disjoint subset
  3. Disjoint bilinear programming and the min-max linear problem with interdependent variables
  4. Estimation of the optimal value of the objective function for the disjoint bilinear programming problem (\ref{['eq1']})-(\ref{['eq3']})
  5. An optimality criterion for disjoint bilinear programming (\ref{['eq1']})-(\ref{['eq3']})
  6. The approach for solving the problem based on Theorem \ref{['tm01']}
  7. Some auxiliary results related to redundant inequalities for linear systems
  8. The optimality criterion for the disjoint bilinear programming problem with a perfect disjoint subset
  9. The properties of the extreme points for the set of solutions of system (\ref{['eq03']})
  10. A criterion for checking if $Y \not \subset Y_h$ based on properties of redundant inequalities for inconsistent systems of linear inequalities
  11. The optimality criterion for the disjoint bilinear programming problem (\ref{['eq01']})-(\ref{['eq03']})
  12. The main results concerned with checking the conditions of Theorem \ref{['thm101']}
  13. Some preliminary results for the problem of checking if $Y \not \subset Y_h$
  14. The main results for checking if $Y\not\subset Y_h$
  15. Polynomial algorithm to check if the condition $Y\not\subset Y_h$ holds for a given $h\in [-2^L,2^L]$
  16. ...and 5 more sections

Key Result

Theorem 1

If $(x^*,y^*)$ is an optimal solution of problem (eq1)-(eq3) and $z^*$ is the optimal value of the objective function in this problem, then $z^*$ and $y^*\in Y$ represent a solution of min-max problem (neweq4)-(neweq6) and vice versa: if $z^*$ and $y^*\in Y$ represent a solution of min-max problem (

Theorems & Definitions (64)

  • Theorem 1
  • proof
  • Theorem 2
  • Corollary 1
  • Lemma 1
  • proof
  • Remark 1
  • Lemma 2
  • proof
  • Corollary 2
  • ...and 54 more