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Title: Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution

Abstract

In order to express a polyhedron as the Minkowski sum of a polytope and a polyhedral cone, Motzkin (Beiträge zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) devised a homogenization technique that translates the polyhedron to a polyhedral cone in one higher dimension. Refining his technique, we present a conical representation of a set in the Euclidean space. Then, we use this representation to reach four main results: First, we establish a convex programming based framework for determining a maximal element—an element with the maximum number of positive components—of a non-negative convex set—a convex set in the non-negative Euclidean orthant. Second, we develop a linear programming problem for finding a relative interior point of a polyhedron. Third, we propose two procedures for identifying a strictly complementary solution in linear programming. Finally, we generalize Motzkin’s (Beiträge zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) representation theorem for a class of closed convex sets in the Euclidean space.

Authors:
 [1];  [2]; ;  [1]
  1. Shiraz University, Department of Mathematics, College of Sciences (Iran, Islamic Republic of)
  2. National Graduate Institute for Policy Studies (Japan)
Publication Date:
OSTI Identifier:
22769394
Resource Type:
Journal Article
Journal Name:
Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 37; Journal Issue: 1; Other Information: Copyright (c) 2018 SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0101-8205
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICAL METHODS AND COMPUTING; EUCLIDEAN SPACE; LINEAR PROGRAMMING; MATHEMATICAL SOLUTIONS

Citation Formats

Mehdiloozad, Mahmood, Tone, Kaoru, Askarpour, Rahim, and Ahmadi, Mohammad Bagher. Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution. United States: N. p., 2018. Web. doi:10.1007/S40314-016-0324-X.
Mehdiloozad, Mahmood, Tone, Kaoru, Askarpour, Rahim, & Ahmadi, Mohammad Bagher. Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution. United States. doi:10.1007/S40314-016-0324-X.
Mehdiloozad, Mahmood, Tone, Kaoru, Askarpour, Rahim, and Ahmadi, Mohammad Bagher. Thu . "Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution". United States. doi:10.1007/S40314-016-0324-X.
@article{osti_22769394,
title = {Finding a maximal element of a non-negative convex set through its characteristic cone: an application to finding a strictly complementary solution},
author = {Mehdiloozad, Mahmood and Tone, Kaoru and Askarpour, Rahim and Ahmadi, Mohammad Bagher},
abstractNote = {In order to express a polyhedron as the Minkowski sum of a polytope and a polyhedral cone, Motzkin (Beiträge zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) devised a homogenization technique that translates the polyhedron to a polyhedral cone in one higher dimension. Refining his technique, we present a conical representation of a set in the Euclidean space. Then, we use this representation to reach four main results: First, we establish a convex programming based framework for determining a maximal element—an element with the maximum number of positive components—of a non-negative convex set—a convex set in the non-negative Euclidean orthant. Second, we develop a linear programming problem for finding a relative interior point of a polyhedron. Third, we propose two procedures for identifying a strictly complementary solution in linear programming. Finally, we generalize Motzkin’s (Beiträge zur Theorie der linearen Ungleichungen. Dissertation, University of Basel, 1936) representation theorem for a class of closed convex sets in the Euclidean space.},
doi = {10.1007/S40314-016-0324-X},
journal = {Computational and Applied Mathematics},
issn = {0101-8205},
number = 1,
volume = 37,
place = {United States},
year = {2018},
month = {3}
}