Quasiconvex Programming

We define quasiconvex programming, a form of generalized linear programming in which one seeks the point minimizing the pointwise maximum of a collection of quasiconvex functions. We survey algorithms for solving quasiconvex programs either numerically or via generalizations of the dual simplex method from linear programming, and describe varied applications of this geometric optimization technique in meshing, scientific computation, information visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure

Set containment characterization and mathematical programming

Recently, many researchers studied set containment characterizations. In this paper, we introduce some set containment characterizations for quasiconvex programming. Furthermore, we show a duality theorem for quasiconvex programming by using set containment characterizations. Notions of quasiconjugate for quasiconvex functions, especially 1, -1-quasiconjugate, 1-semiconjugate, H-quasiconjugate and R-quasiconjugate, play important roles to derive characterizations of the set containments

Optimization problems with quasiconvex inequality constraints

The constrained optimization problem min f(x), gj(x) 0 (j = 1, . . . , p) is considered, where f : X ! R and gj : X ! R are nonsmooth functions with domain X Rn. First-order necessary and first-order sufficient optimality conditions are obtained when gj are quasiconvex functions. Two are the main features of the paper: to treat nonsmooth problems it makes use of the Dini derivative; to obtain more sensitive conditions, it admits directionally dependent multipliers. The two cases, where the Lagrange function satisfies a non-strict and a strict inequality, are considered. In the case of a non-strict inequality pseudoconvex functions are involved and in their terms some properties of the convex programming problems are generalized. The efficiency of the obtained conditions is illustrated on an example. Key words: Nonsmooth optimization, Dini directional derivatives, quasiconvex functions, pseudoconvex functions, quasiconvex programming, Kuhn-Tucker conditions.

Optimality conditions for scalar and vector optimization problems with quasiconvex inequality constraints

Let X be a real linear space, X0 X a convex set, Y and Z topological real linear spaces. The constrained optimization problem minCf(x), g(x) 2 -K is considered, where f : X0 ! Y and g : X0 ! Z are given (nonsmooth) functions, and C Y and K Z are closed convex cones. The weakly efficient solutions (w-minimizers) of this problem are investigated. When g obeys quasiconvex properties, first-order necessary and first-order sufficient optimality conditions in terms of Dini directional derivatives are obtained. In the special case of problems with pseudoconvex data it is shown that these conditions characterize the global w-minimizers and generalize known results from convex vector programming. The obtained results are applied to the special case of problems with finite dimensional image spaces and ordering cones the positive orthants, in particular to scalar problems with quasiconvex constraints. It is shown, that the quasiconvexity of the constraints allows to formulate the optimality conditions using the more simple single valued Dini derivatives instead of the set valued ones. Key words: Vector optimization, nonsmooth optimization, quasiconvex vector functions, pseudoconvex vector functions, Dini derivatives, quasiconvex programming, Kuhn-Tucker conditions..

On the Necessity of the Sufficient Conditions in Cone-Constrained Vector Optimization

The object of investigation in this paper are vector nonlinear programming problems with cone constraints. We introduce the notion of a Fritz John pseudoinvex cone-constrained vector problem. We prove that a problem with cone constraints is Fritz John pseudoinvex if and only if every vector critical point of Fritz John type is a weak global minimizer. Thus, we generalize several results, where the Paretian case have been studied. We also introduce a new Frechet differentiable pseudoconvex problem. We derive that a problem with quasiconvex vector-valued data is pseudoconvex if and only if every Fritz John vector critical point is a weakly efficient global solution. Thus, we generalize a lot of previous optimality conditions, concerning the scalar case and the multiobjective Paretian one. Additionally, we prove that a quasiconvex vector-valued function is pseudoconvex with respect to the same cone if and only if every vector critical point of the function is a weak global minimizer, a result, which is a natural extension of a known characterization of pseudoconvex scalar functions.Comment: 12 page