The aim of this paper is to give a survey of some basic theory of semi-infinite programming. In particular, we discuss various approaches to derivations of duality, discretization and first and second order optimality conditions. Some of the surveyed results are well known while others seem to be less noticed in that area of research. … Read more
In this work, we develop duality of the minimum volume circumscribed ellipsoid and the maximum volume inscribed ellipsoid problems. We present a unified treatment of both problems using convex semi–infinite programming. We establish the known duality relationship between the minimum volume circumscribed ellipsoid problem and the optimal experimental design problem in statistics. The duality results … Read more
In many path planning situations we would like to find a path of constrained Euclidean length in 2D that minimises a line integral. We call this the Continuous Length-Constrained Minimum Cost Path Problem (C-LCMCPP). Generally, this will be a non-convex optimization problem, for which continuous approaches only ensure locally optimal solutions. However, network discretisations yield … Read more
A convex body K has associated with it a unique circumscribed ellipsoid CE(K) with minimum volume, and a unique inscribed ellipsoid IE(K) with maximum volume. We first give a unified, modern exposition of the basic theory of these extremal ellipsoids using the semi-infinite programming approach pioneered by Fritz John in his seminal 1948 paper. We … Read more
Robust optimization programs are hard to solve even when the constraints are convex. In previous contributions, it has been shown that approximately robust solutions (i.e. solutions feasible for all constraints but a small fraction of them) to convex programs can be obtained at low computational cost through constraints randomization. In this paper, we establish new … Read more
We propose a novel optimization model for risk-averse investors to obtain robust solutions for portfolio allocation problems. Unlike related models in the literature, no historical data or statistical estimation techniques are used to compute the parameters of the model. Instead, the parameters are directly obtained from current prices of options on the assets being considered. … Read more
This paper provides sufficient conditions for the optimal value function of a given linear semi-infinite programming problem to depend linearly on the size of the perturbations, when these perturbations are directional, involve either the cost coefficients or the right-hand-side function or both, and they are sufficiently small. Two kinds of partitions are considered. The first … Read more
It is well known that the robust counterpart introduced by Ben-Tal and Nemirovski [2] increases the numerical complexity of the solution compared to the original problem. Kocvara, Nemirovski and Zowe therefore introduced in [9] an approximation algorithm for the special case of robust material optimization, called cascading. As the title already indicates, we will show … Read more
We show that the problem of computing sharp upper and lower static-arbitrage bounds on the price of a European basket option, given the prices of other similar options, can be cast as a linear program (LP). The LP formulations readily yield super-replicating (sub-replicating) strategies for the upper (lower) bound problem. The dual counterparts of the … Read more
Based on previous explicit computations of universal barrier functions, we describe numerical experiments for solving certain classes of convex optimization problems. The comparison is given of the performance of the classical affine-scaling algorithm with similar algorithm based upon the universal barrier function Citation To appear in “Computational Optimization and Applications” Article Download View Numerical Experiments … Read more