Optimization and Equilibrium Problems with Equilibrium Constraints

The paper concerns optimization and equilibrium problems with the so-called equilibrium constraints (MPEC and EPEC), which frequently appear in applications to operations research. These classes of problems can be naturally unified in the framework of multiobjective optimization with constraints governed by parametric variational systems (generalized equations, variational inequalities, complementarity problems, etc.). We focus on necessary conditions for optimal solutions to MPECs and EPECs under general assumptions in finite-dimensional spaces. Since such problems are intrinsically nonsmooth, we use advanced tools of generalized differentiation to study optimal solutions by methods of modern variational analysis. The general results obtained are concretized for special classes of MPECs and EPECs important in applications

A MPCC-NLP approach for an electric power market problem

The electric power market is changing - it has passed from a regulated market, where the government of each country had the control of prices, to a deregulated market economy. Each company competes in order to get more clients and maximize its profits. This market is represented by a Stackelberg game with two firms, leader and follower, and the leader anticipates the reaction of the follower. The problem is formulated as a Mathematical Program with Complementarity Constraints (MPCC). It is shown that the constraint qualifications usually assumed to prove convergence of standard algorithms fail to hold for MPCC. To circumvent this, a reformulation for a nonlinear problem (NLP) is proposed. Numerical tests using the NEOS server platform are presented

On M-stationary points for a stochastic equilibrium problem under equilibrium constraints in electricity spot market modeling

Modeling several competitive leaders and followers acting in an electricity marketleads to coupled systems of mathematical programs with equilibrium constraints,called equilibrium problems with equilibrium constraints (EPECs). We consider asimplified model for competition in electricity markets under uncertainty of demandin an electricity network as a (stochastic) multi-leader-follower game. First ordernecessary conditions are developed for the corresponding stochastic EPEC based ona result of Outrata [17]. For applying the general result an explicit representation ofthe co-derivative of the normal cone mapping to a polyhedron is derived (Proposition3.2). Later the co-derivative formula is used for verifying constraint qualificationsand for identifying M-stationary solutions of the stochastic EPEC if the demand isrepresented by a finite number of scenarios

Risk, Security and Robust Solutions

The aim of this paper is to develop a decision-theoretic approach to security management of uncertain multi-agent systems. Security is defined as the ability to deal with intentional and unintentional threats generated by agents. The main concern of the paper is the protection of public goods from these threats allowing explicit treatment of inherent uncertainties and robust security management solutions. The paper shows that robust solutions can be properly designed by new stochastic optimization tools applicable for multicriteria problems with uncertain probability distributions and multivariate extreme events

Artificial Intelligence Techniques for Automatic Reformulation and Solution of Structured Mathematical Models

Complex, hierarchical, multi-scale industrial and natural systems generate increasingly large mathematical models. Practitioners are usually able to formulate such models in their "natural" form; however, solving them often requires finding an appropriate reformulation to reveal structures in the model which make it possible to apply efficient, specialized approaches. The search for the "best" formulation of a given problem, the one which allows the application of the solution algorithm that best exploits the available computational resources, is currently a painstaking process which requires considerable work by highly skilled personnel. Experts in solution algorithms are required for figuring out which (formulation, algorithm) pair is better used, considering issues like the appropriate selection of the several obscure algorithmic parameters that each solution methods has. This process is only going to get more complex, as current trends in computer technology dictate the necessity to develop complex parallel approaches capable of harnessing the power of thousands of processing units, thereby adding another layer of complexity in the form of the choice of the appropriate (parallel) architecture. All this renders the use of mathematical models exceedingly costly and difficult for many potentially fruitful applications. The \name{} environment, proposed in this Thesis, aims at devising a software system for automatizing the search for the best combination of (re)formulation, solution algorithm and its parameters (comprised the computational architecture), until now a firm domain of human intervention, to help practitioners bridging the gap between mathematical models cast in their natural form and existing solver systems. I-DARE deals with deep and challenging issues, both from the theoretical and from an implementative viewpoint: 1) the development of a language that can be effectively used to formulate large-scale structured mathematical models and the reformulation rules that allow to transform a formulation into a different one; 2) a core subsystem capable of automatically reformulating the models and searching in the space of (formulations, algorithms, configurations) able to "the best" formulation of a given problem; 3) the design of a general interface for numerical solvers that is capable of accommodate and exploit structure information. To achieve these goals I-DARE will propose a sound and articulated integration of different programming paradigms and techniques like, classic Object-Oriented programing and Artificial Intelligence (Declarative Programming, Frame-Logic, Higher-Order Logic, Machine Learning). By tackling these challenges, I-DARE may have profound, lasting and disruptive effects on many facets of the development and deployment of mathematical models and the corresponding solution algorithms