# 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:

- Shiraz University, Department of Mathematics, College of Sciences (Iran, Islamic Republic of)
- 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}

}