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# Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming

## Author

Listed:
• A. L. Soyster

(Pennsylvania State University, University Park, Pennsylvania)

## Abstract

This note formulates a convex mathematical programming problem in which the usual definition of the feasible region is replaced by a significantly different strategy. Instead of specifying the feasible region by a set of convex inequalities, f i ( x ) ≦ b i , i = 1, 2, …, m , the feasible region is defined via set containment. Here n convex activity sets { K j , j = 1, 2, …, n } and a convex resource set K are specified and the feasible region is given by \documentclass{aastex}\usepackage{amsbsy}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{bm}\usepackage{mathrsfs}\usepackage{pifont}\usepackage{stmaryrd}\usepackage{textcomp}\usepackage{portland,xspace}\usepackage{amsmath,amsxtra}\pagestyle{empty}\DeclareMathSizes{10}{9}{7}{6}\begin{document}$$X =\{x \in R^{n}\mid x_{1}K_{1} + x_{2}K_{2} + \cdots + x_{n}K_{n} \subseteq K, x_{j}\geq 0\}$$\end{document} where the binary operation + refers to addition of sets. The problem is then to find x̄ ∈ X that maximizes the linear function c · x . When the resource set has a special form, this problem is solved via an auxiliary linear-programming problem and application to inexact linear programming is possible.

## Suggested Citation

• A. L. Soyster, 1973. "Technical Note—Convex Programming with Set-Inclusive Constraints and Applications to Inexact Linear Programming," Operations Research, INFORMS, vol. 21(5), pages 1154-1157, October.
• Handle: RePEc:inm:oropre:v:21:y:1973:i:5:p:1154-1157
DOI: 10.1287/opre.21.5.1154
as

File URL: http://dx.doi.org/10.1287/opre.21.5.1154

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