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This paper introduces novel numerical solution strategies for generalized semi-infinite optimization problems (GSIP), a class of mathematical optimization problems which occur naturally in the context of design centering problems, robust optimization problems, and many fields of engineering science. GSIPs can be regarded as bilevel optimization problems, where a parametric lower-level maximization problem has to be solved in order to check feasibility of the upper level minimization problem. The current paper discusses several strategies to reformulate this class of problems into equivalent finite minimization problems by exploiting the concept of Wolfe duality for convex lower level problems. Here, the main contribution is the discussion of the non-degeneracy of the corresponding formulations under various assumptions. Finally, these non-degenerate reformulations of the original GSIP allow us to apply standard nonlinear optimization algorithms. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to give a survey of methods, partly derived by the author in joint work with other researchers, concerning the problem of constructing e-optimal strategies for partially observable MDPs. The methods basically consist in transforming the problem into one of approximation: Starting from the original problem a sequence of approximating problems is constructed such that: (i) For each approximating problem an optimal strategy can actually be computed, (ii) Given ϵ » 0, there exists an approximating problem such that the optimal strategy for the latter is ϵ-optimal for the original problem. [ABSTRACT FROM AUTHOR]

The paper suggests a new-to the best of the author's knowledge-characterization of decisions, which are optimal in the multi-objective optimization problem with respect to a definite proper preference cone, a Euclidean cone with a prescribed angular radius. The main idea is to use the angle distances between the unit vector and points of utility space. A necessary and sufficient condition for the optimality in the form of an equation is derived. The first-order necessary optimality conditions are also obtained. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a definition of generalized convexlike functions (preconvexlike functions). Then, under the weakened convexity, we study vector optimization problems in Hausdorff topological linear spaces. We establish some generalized Motzkin theorems of the alternative. By use of these theorems of the alternative, we obtain some Lagrangian multiplier theorems. A saddle-point theorem and a scalarization theorem are also derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the relationship between bilevel optimization and multicriteria optimization. Given a bilevel optimization problem, we introduce an order relation such that the optimal solutions of the bilevel problem are the nondominated points with respect to the order relation. In the case where the lower-level problem of the bilevel optimization problem is convex and continuously differentiable in the lower-level variables, this order relation is equivalent to a second, more tractable order relation. Then, we show how to construct a (nonconvex) cone for which we can prove that the nondominated points with respect to the order relation induced by the cone are also nondominated points with respect to any of the two order relations mentioned before. We comment also on the practical and computational implications of our approach. [ABSTRACT FROM AUTHOR]

Two basic disadvantages of the symmetric rank one (SR1) update are that the SR1 update may not preserve positive definiteness when starting with a positive definite approximation and the SR1 update can be undefined. A simple remedy to these problems is to restart the update with the initial approximation, mostly the identity matrix, whenever these difficulties arise. However, numerical experience shows that restart with the identity matrix is not a good choice. Instead of using the identity matrix we used a positive multiple of the identity matrix. The used positive scaling factor is the optimal solution of the measure defined by the problem—maximize the determinant of the update subject to a bound of one on the largest eigenvalue. This measure is motivated by considering the volume of the symmetric difference of the two ellipsoids, which arise from the current and updated quadratic models in quasi-Newton methods. A replacement in the form of a positive multiple of the identity matrix is provided for the SR1 update when it is not positive definite or undefined. Our experiments indicate that with such simple initial scaling the possibility of an undefined update or the loss of positive definiteness for the SR1 method is avoided on all iterations. [ABSTRACT FROM AUTHOR]

Traveling Salesman, Steiner Tree, and many other famous geometric optimization problems are NP-hard. Since we do not expect to design efficient algorithms that solve these problems optimally, researchers have tried to design approximation algorithms, which can compute a provably near-optimal solution in polynomial time.We survey such algorithms, in particular a new technique developed over the past few years that allows us to design approximation schemes for many of these problems. For any fixed constant c> 0, the algorithm can compute a solution whose cost is at most (1 + c) times the optimum. (The running time is polynomial for every fixed c> 0, and in many cases is even nearly linear.) We describe how these schemes are designed, and survey the status of a large number of problems. [ABSTRACT FROM AUTHOR]