On the behavior of Lagrange multipliers in convex and non-convex infeasible interior point methods

This paper analyzes sequences generated by infeasible interior point methods. In convex and non-convex settings, we prove that moving the primal feasibility at the same rate as complementarity will ensure that the Lagrange multiplier sequence will remain bounded, provided the limit point of the primal sequence has a Lagrange multiplier, without constraint qualification assumptions. We … Read more

A second-order optimality condition with first- and second-order complementarity associated with global convergence of algorithms

We develop a new notion of second-order complementarity with respect to the tangent subspace related to second-order necessary optimality conditions by the introduction of so-called tangent multipliers. We prove that around a local minimizer, a second-order stationarity residual can be driven to zero while controlling the growth of Lagrange multipliers and tangent multipliers, which gives … Read more

An approximation scheme for a class of risk-averse stochastic equilibrium problems

We consider two models for stochastic equilibrium: one based on the variational equilibrium of a generalized Nash game, and the other on the mixed complementarity formulation. Each agent in the market solves a one-stage risk-averse optimization problem with both here-and-now (investment) variables and (production) wait-and-see variables. A shared constraint couples almost surely the wait-and-see decisions … Read more

Branch-and-Cut for Complementarity-Constrained Optimization

We report and analyze the results of our computational testing of branch-and-cut for the complementarity-constrained optimization problem (CCOP). Besides the MIP cuts commonly present in commercial optimization software, we used inequalities that explore complementarity constraints. To do so, we generalized two families of cuts proposed earlier by de Farias, Johnson, and Nemhauser that had never … Read more

Real-Time Optimization as a Generalized Equation

We establish results for the problem of tracking a time-dependent manifold arising in online nonlinear programming by casting this as a generalized equation. We demonstrate that if points along a solution manifold are consistently strongly regular, it is possible to track the manifold approximately by solving a linear complementarity problem (LCP) at each time step. … Read more

Formulation of Oligopolistic Competition in AC Power Networks: An NLP Approach

In this paper, oligopolistic competition in a centralized power market is characterized by a multi-leader single-follower game, and formulated as a nonlinear programming (NLP) problem. An AC network is used to represent the transmission system and is modeled using rectangular coordinates. The follower is composed of a set of competitive suppliers, demands, and the system … Read more

Numerical Study of Affine Supply Function Equilibrium in AC Network-Constrained Markets

An affine supply function equilibrium (SFE) approach is used to discuss voltage constraints and reactive power issues in the modeling of strategic behavior. Generation companies (GenCos) can choose their bid parameters with no restrictions for both energy and spinning reserves. The strategic behavior of generators is formulated as a multi-leader single-follower game. Each GenCo is … Read more

Convergence Analysis of an Interior-Point Method for Mathematical Programs with Equilibrium Constraints

We prove local and global convergence results for an interior-point method applied to mathematical programs with equilibrium constraints. The global result shows the algorithm minimizes infeasibility regardless of starting point, while one result proves local convergence when penalty functions are exact; another local result proves convergence when the solution is not even a KKT point. … Read more

The Q Method for Symmetric Cone Programming

We extend the Q method to the symmetric cone programming. An infeasible interior point algorithm and a Newton-type algorithm are given. We give convergence results of the interior point algorithm and prove that the Newton-type algorithm is good for Citation AdvOl-Report#2004/18 McMaster University, Advanced Optimization Laboratory Hamilton, Ontario, Canada October 2004 Article Download View The … Read more

An Algorithm for Perturbed Second-order Cone Programs

The second-order cone programming problem is reformulated into several new systems of nonlinear equations. Assume the perturbation of the data is in a certain neighborhood of zero. Then starting from a solution to the old problem, the semismooth Newton’s iterates converge Q-quadratically to a solution of the perturbed problem. The algorithm is globalized. Numerical examples … Read more