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

Journal Article · Thu Mar 15 04:00:00 EDT 2018 · Computational and Applied Mathematics

DOI:https://doi.org/10.1007/S40314-016-0324-X· OSTI ID:22769394

Mehdiloozad, Mahmood ^[1]; Tone, Kaoru ^[2]; Askarpour, Rahim; Ahmadi, Mohammad Bagher ^[1]

---

1. Shiraz University, Department of Mathematics, College of Sciences (Iran, Islamic Republic of)
2. National Graduate Institute for Policy Studies (Japan)

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.

• OSTI ID: 22769394
• Journal Information: Computational and Applied Mathematics, Journal Name: Computational and Applied Mathematics Journal Issue: 1 Vol. 37; ISSN 0101-8205
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
EUCLIDEAN SPACE
LINEAR PROGRAMMING
MATHEMATICAL SOLUTIONS