This paper presents a Nash equilibrium model where the underlying objective functions involve uncertainty and nonsmoothness. The well known sample average approximation method is applied to solve the problem and the first order equilibrium conditions are characterized in terms of Clarke generalized gradients. Under some moderate conditions, it is shown that with probability one, a … Read more
Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body~$\mathfrak x$, we try to maximize the volume of~$\mathfrak x$ and minimize the width of~$\mathfrak x$ simultaneously. These problems are addressed … Read more
We consider a quadratic optimal control problem governed by a nonautonomous affine differential equation subject to nonnegativity control constraints. For a general class of interior penalty functions, we show how to compute the principal term of the pointwise expansion of the state and the adjoint state. Our main argument relies on the following fact: If … Read more
In this paper, we present an Uzawa-based heuristic that is adapted to some type of stochastic optimal control problems. More precisely, we consider dynamical systems that can be divided into small-scale independent subsystems, though linked through a static almost sure coupling constraint at each time step. This type of problem is common in production/portfolio management … Read more
Several efficient methods for derivative-free optimization (DFO) are based on the construction and maintenance of an interpolation model for the objective function. Most of these algorithms use special “geometry-improving” iterations, where the geometry (poisedness) of the underlying interpolation set is made better at the cost of one or more function evaluations. We show that such … Read more
We consider risk-averse formulations of stochastic linear programs having a structure that is common in real-life applications. Specifically, the optimization problem corresponds to controlling over a certain horizon a system whose dynamics is given by a transition equation depending affinely on an interstage dependent stochastic process. We put in place a rolling-horizon time consistent policy. … Read more
We present a proximal point method to solve multiobjective problems based on the scalarization for maps. We build a family of a convex scalar strict representation of a convex map F with respect to the lexicographic order on Rm and we add a variant of the logarithmquadratic regularization of Auslender, where the unconstrained variables in … Read more
A simple unified framework is presented for the study of strong efficient solutions, weak efficient solutions, positive proper efficient solutions, Henig global proper efficient solutions, Henig proper efficient solutions, super efficient solutions, Benson proper efficient solutions, Hartley proper efficient solutions, Hurwicz proper efficient solutions and Borwein proper efficient solutions of set-valued optimization problem with/or without … Read more
In this paper we present a primal-dual interior-point algorithm to solve a class of multi-objective network flow problems. More precisely, our algorithm is an extension of the single-objective primal-dual infeasible and inexact interior point method for multi-objective linear network flow problems. A comparison with standard interior point methods is provided and experimental results on bi-objective … Read more
In this paper we consider, for the first time, approximate Henig proper minimizers and approximate super minimizers of a set-valued map F with values in a partially ordered vector space and formulate two versions of the Ekeland variational principle for these points involving coderivatives in the senses of Ioffe, Clarke and Mordukhovich. As applications we … Read more