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On calculating the normal cone to a finite union of convex polyhedra
The paper provides formulae for calculating the limiting normal cone introduced by Mordukhovich to a finite union of convex polyhedra. In the first part, special cases of independent interest are considered (almost disjoint cones, half spaces, orthants). The second part focusses on unions of general polyhedra. Due to the local nature of the normal cone, one may restrict considerations without loss of generality to finite unions of polyhedral cones. First, an explicit formula for the normal cone is provided in the situation of two cones. An algorithmic approach is presented along with a refined, more efficient formula. Afterwards, a general formula for the union of N cones is derived. Finally, an application to the stability analysis of a special type of probabilistic constraints is provided.
Solving joint chance constrained problems using regularization and Benders decomposition
In this paper we investigate stochastic programms with joint chance constraints. We consider discrete scenario set and reformulate the problem by adding auxiliary variables. Since the resulting problem has a difficult feasible set, we regularize it. To decrease the dependence on the scenario number, we propose a numerical method by iteratively solving a master problem while adding Benders cuts. We find the solution of the slave problem (generating the Benders cuts) in a closed form and propose a heuristic method to decrease the number of cuts. We perform a numerical study by increasing the number of scenarios and compare our solution with a solution obtained by solving the same problem with continuous distribution.
A joint model of probabilistic/robust constraints for gas transport management in stationary networks
We present a novel mathematical algorithm to assist gas network operators in managing uncertainty, while increasing reliability of transmission and supply. As a result, we solve an optimization problem with a joint probabilistic constraint over an infinite system of random inequalities. Such models arise in the presence of uncertain parameters having partially stochastic and partially nonstochastic character. The application that drives this new approach is a stationary network with uncertain demand (which are stochastic due to the possibility of fitting statistical distributions based on historical measurements) and with uncertain roughness coefficients in the pipes (which are uncertain but non-stochastic due to a lack of attainable measurements). We study the sensitivity of local uncertainties in the roughness coefficients and their impact on a highly reliable network operation. In particular, we are going to answer the question, what is the maximum uncertainty that is allowed (shaping a maximal uncertainty set) around nominal roughness coefficients, such that random demands in a stationary gas network can be satisfied at given high probability level for no matter which realization of true roughness coefficients within the uncertainty set. One ends up with a constraint, which is probabilistic with respect to the load of gas and robust with respect to the roughness coefficients. We demonstrate how such constraints can be dealt with in the framework of the so-called spheric-radial decomposition of multivariate Gaussian distributions. The numerical solution of a corresponding optimization problem is illustrated. The results might assist the network operator with the implementation of cost-intensive roughness measurements.
Inradius and circumradius of various convex cones arising in applications
This note addresses the issue of computing the inradius and the circumradius of a convex cone in a Euclidean space. It deals also with the related problem of finding the incenter and the circumcenter of the cone. We work out various examples of convex cones arising in applications.
Outer limit of subdifferentials and calmness moduli in linear and nonlinear programming
With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a KKT index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.
Condition number and eccentricity of a closed convex cone
We discuss some extremality issues concerning the circumradius, the inradius, and the condition number of a closed convex cone in $mathbbR^n$. The condition number refers to the ratio between the circumradius and the inradius. We also study the eccentricity of a closed convex cone, which is a coefficient that measures to which extent the circumcenter differs from the incenter.
On properties of different notions of centers for convex cones
The points on the revolution axis of a circular cone are somewhat special: they are the "most interior'' elements of the cone. This paper addresses the issue of formalizing the concept of center for a convex cone that is not circular. Four distinct proposals are studied in detail: the incenter, the circumcenter, the inner center, and the outer center. The discussion takes place in the context of a reflexive Banach space.
Uniform boundedness of norms of convex and nonconvex processes
The lower limit of a sequence of closed convex processes is again a closed convex process. In this note we prove the following uniform boundedness principle: if the lower limit is nonempty-valued everywhere, then, starting from a certain index, the given sequence is uniformly norm-bounded. As shown with an example, the uniform boundedness principle is not true if one drops convexity. By way of illustration, we consider an application to the controllability analysis of differential inclusions.
Chance constraints in PDE constrained optimization
Chance constraints represent a popular tool for finding decisions that enforce a robust satisfaction of random inequality systems in terms of probability. They are widely used in optimization problems subject to uncertain parameters as they arise in many engineering applications. Most structural results of chance constraints (e.g., closedness, convexity, Lipschitz continuity, differentiability etc.) have been formulated in a finite-dimensional setting. The aim of this paper is to generalize some of these well-known semi-continuity and convexity properties to a setting of control problems subject to (uniform) state chance constraints.
On calmness conditions in convex bilevel programming
In this article we compare two different calmness conditions which are widely used in the literature on bilevel programming and on mathematical programs with equilibrium constraints. In order to do so, we consider convex bilevel programming as a kind of intersection between both research areas. The so-called partial calmness concept is based on the function value approach for describing the lower level solution set. Alternatively, calmness in the sense of multifunctions may be considered for perturbations of the generalized equation representing the same lower level solution set. Both concepts allow to derive first order necessary optimality conditions via tools of generalized differentiation introduced by Mordukhovich. They are very different, however, concerning their range of applicability and the form of optimality conditions obtained. The results of this paper seem to suggest that partial calmness is considerably more restrictive than calmness of the perturbed generalized equation. This fact is also illustrated by means of a dicretized obstacle control problem.