41 research outputs found
Generalized semi-infinite programming: Numerical aspects
Generalized semi-infinite optimization problems (GSIP) are considered. It is investigated how the numerical methods for standard semi-infinite programming (SIP) can be extended to GSIP. Newton methods can be extended immediately. For discretization methods the situation is more complicated. These difficulties are discussed and convergence results for a discretization and an exchange method are derived under fairly general assumptions. The question under which conditions GSIP represents a convex problem is answered
How to split the eigenvalues of a one-parameter family of matrices
We are concerned with families of -matrices depending smoothly on the parameter . We survey results on the behaviour of eigenvalues of for certain classes of matrices. We are especially interested in the question whether multiple eigenvalues can be avoided generically. In the set of families of symmetric matrices , for example, generically all eigenvalues of are simple for all . We consider a class of natural perturbations of a given matrix family such that lies in the generic class, i.e.\ avoids double eigenvalues `as far as possible'
An easy way to obtain strong duality results in linear, linear semidefinite and linear semi-infinite programming
In linear programming it is known that an appropriate non-homogeneous Farkas Lemma leads to a short proof of the strong duality results for a pair of primal and dual programs. By using a corresponding generalized Farkas lemma we give a similar proof of the strong duality results for semidefinite programs under constraint qualifications. The proof includes optimality conditions. The same approach leads to corresponding results for linear semi-infinite programs. For completeness, the proofs for linear programs and the proofs of all auxiliary lemmata for the semidefinite case are included
Linear bilevel problems: Genericity results and an efficient method for computing local minima
The paper is concerned with linear bilevel problems. These nonconvex problems are known to be NP-complete. So, no efficient method for solving the global bilevel problem can be expected. In this paper we give a genericity analysis of linear bilevel problems and present a new algorithm for computing efficiently local minimizers. The method is based on the given structural analysis and combines ideas of the Simplex method with projected gradient steps
On generalized semi-infinite optimization and bilevel optimization
The paper studies the connections and differences between bilevel problems (BL) and generalized semi-infinite problems (GSIP). Under natural assumptions (GSIP) can be seen as a special case of a (BL). We consider the so-called reduction approach for (BL) and (GSIP) leading to optimality conditions and Newton-type methods for solving the problems. We show by a structural analysis that for (GSIP)-problems the regularity assumptions for the reduction approach can be expected to hold generically at a solution but for general (BL)-problems not. The genericity behavior of (BL) and (GSIP) is in particular studied for linear problems
The generalized minimum spanning tree problem
We consider the Generalized Minimum Spanning Tree Problem denoted by GMSTP. It is known that GMSTP is NP-hard and even finding a near optimal solution is NP-hard. We introduce a new mixed integer programming formulation of the problem which contains a polynomial number of constraints and a polynomial number of variables. Based on this formulation we give an heuristic solution, a lower bound procedure and an upper bound procedure and present the advantages of our approach in comparison with an earlier method. We present a solution procedure for solving GMST problem using cutting planes
An approximation algorithm for the generalized minimum spanning tree problem with bounded cluster size
Given a complete undirected graph with the nodes partitioned into m node sets called clusters, the Generalized Minimum Spanning Tree problem denoted by GMST is to find a minimum-cost tree which includes exactly one node from each cluster. It is known that the GMST problem is NP-hard and even finding a near optimal solution is NP-hard. We give an approximation algorithm for the Generalized Minimum Spanning Tree problem in the case when the cluster size is bounded by . In this case, the GMST problem can be approximated to within 2
Discretization in semi-infinite programming: The rate of approximation
The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the rate of the approximation error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size . It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a consistent way. This is done for common and for generalized semi-infinite problems
An elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions in nonlinear programming
In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints.The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization ofFarkas lemma and the Bolzano-Weierstrass property for compact sets.Fritz-John conditions;Karush-Kuhn-Tucker conditions;nonlinear programming
A Note on the Dual of an Unconstrained (Generalized) Geometric Programming Problem
In this note we show that the strong duality theorem of an unconstrained (generalized) geometric
programming problem as defined by Peterson (cf.[1]) is actually a special case of a Lagrangian
duality result. Contrary to [1] we also consider the case that the set C is compact and convex
and in this case we do not need to assume the standard regularity condition