Variational analysis and full stability of optimal solutions to constrained and minimax problems

Categories Convex and Nonsmooth Optimization

The main goal of this paper is to develop applications of advanced tools of first-order and second-order variational analysis and generalized differentiation to the fundamental notion of full stability of local minimizers of general classes of constrained optimization and minimax problems. In particular, we derive second-order characterizations of full stability and investigate its relationships with … Read more

An induction theorem and nonlinear regularity models

Categories Convex and Nonsmooth Optimization Tags , ,

A general nonlinear regularity model for a set-valued mapping $F:X\times\R_+\rightrightarrows Y$, where $X$ and $Y$ are metric spaces, is considered using special iteration procedures, going back to Banach, Schauder, Lusternik and Graves. Namely, we revise the \emph{induction theorem} from Khanh, \emph{J. Math. Anal. Appl.}, 118 (1986) and employ it to obtain basic estimates for studying … Read more

Data-driven learning in dynamic pricing using adaptive optimization

Categories Applications - OR and Management Sciences, Robust Optimization Tags , , , , , ,

We consider the pricing problem faced by a retailer endowed with a finite inventory of a product offered over a finite planning horizon in an environment where customers are price-sensitive. The parameters of the product demand curve are fixed but unknown to the seller who only has at his disposal a history of sales data. … Read more

PSMG-A Parallel Structured Model Generator for Mathematical Programming

Categories Optimization Software and Modeling Systems, Parallel Algorithms

In this paper, we present PSMG–Parallel Structured Model Generator–an efficient parallel implementation of a model generator for the structure conveying modelling language (SML[4]). Unlike the earlier proof-of-concept implementation presented with SML, PSMG does not depend on AMPL. The main purposes of PSMG are: to provide an easy to use framework for modelling and generating large … Read more

The global weak sharp minima with explicit exponents in polynomial vector optimization problems

Categories Convex and Nonsmooth Optimization, Multi-Criteria Optimization Tags , ,

In this paper we discuss the global weak sharp minima property for vector optimization problems with polynomial data. Exploiting the imposed polynomial structure together with tools of variational analysis and a quantitative version of \L ojasiewicz’s gradient inequality due to D’Acunto and Kurdyka, we establish the H\”older type global weak sharp minima with explicitly calculated … Read more

MILP formulations for the modularity density maximization problem

Categories (Mixed) Integer Linear Programming, Network Optimization

Cluster analysis refers to finding subsets of vertices of a graph (called clusters) which are more likely to be joined pairwise than vertices in different clusters. In the last years this topic has been studied by many researchers, and several methods have been proposed. One of the most popular is to maximize the modularity, which … Read more

Convergence rate analysis of the forward-Douglas-Rachford splitting scheme

Categories Convex and Nonsmooth Optimization Tags , , , , , , ,

Operator splitting schemes are a class of powerful algorithms that solve complicated monotone inclusion and convex optimization problems that are built from many simpler pieces. They give rise to algorithms in which all simple pieces of the decomposition are processed individually. This leads to easily implementable and highly parallelizable or distributed algorithms, which often obtain … Read more

Interior-point solver for convex separable block-angular problems

Categories Convex Optimization, Optimization Software and Modeling Systems

Constraints matrices with block-angular structures are pervasive in Optimization. Interior-point methods have shown to be competitive for these structured problems by exploiting the linear algebra. One of these approaches solved the normal equations using sparse Cholesky factorizations for the block constraints, and a preconditioned conjugate gradient (PCG) for the linking constraints. The preconditioner is based … Read more

A Flexible Iterative Solver for Nonconvex, Equality-Constrained Quadratic Subproblems

Categories Constrained Nonlinear Optimization, Multidisciplinary Design Optimization, Optimization of Systems modeled by PDEs Tags , , , , ,

We present an iterative primal-dual solver for nonconvex equality-constrained quadratic optimization subproblems. The solver constructs the primal and dual trial steps from the subspace generated by the generalized Arnoldi procedure used in flexible GMRES (FGMRES). This permits the use of a wide range of preconditioners for the primal-dual system. In contrast with FGMRES, the proposed … Read more

Decomposition theorems for linear programs

Categories Linear Programming, Network Optimization Tags , , , ,

It is well known that any feasible arc-flow solution to a network problem defined on a graph $G = (N, A)$, where $N$ is the set of nodes whereas $A$ is the set of arcs, can be expressed using at most $|A| + |N|$ paths and cycles having nonzero flow, out of these, at most … Read more