Abstract. In this paper, we present global and local convergence results for an interior-point method for nonlinear programming and analyze the computational performance of its implementation. The algorithm uses an ℓ1 penalty approach to relax all constraints, to provide regularization, and to bound the Lagrange multipliers. The penalty problems are solved using a simplified version of Chen and Goldfarb’s strictly feasible interior-point method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Q-quadratically for a large class of problems. The proposed approach is the first to simultaneously have all of the following properties while solving a general nonconvex nonlinear programming problem: (1) the convergence analysis does not assume boundedness of dual iterates, (2) local convergence does not require the Linear Independence Constraint Qualification, (3) the solution of the penalty problem is shown to locally converge to optima that may not satisfy the Karush-Kuhn-Tucker conditions, and (4) the algorithm is applicable to mathematical programs with equilibrium constraints. Numerical testing on a set of general nonlinear programming problems, including degenerate problems and infeasible problems, confirm the theoretical results. We also provide comparisons to a highly-efficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1.

Abstract. One perceived deficiency of interior-point methods in comparison to active set methods is their inability to efficiently re-optimize by solving closely related problems after a warmstart. In this paper, we investigate the use of a primal-dual penalty approach to overcome this problem. We prove exactness and convergence and show encouraging numerical results on a set of linear and mixed integer programming problems. 1.

Abstract. We propose a relaxation scheme for mathematical programs with equilibrium constraints (MPECs). In contrast to previous approaches, our relaxation is two-sided: both the complementarity and the nonnegativity constraints are relaxed. The proposed relaxation update rule guarantees (under certain conditions) that the sequence of relaxed subproblems will maintain a strictly feasible interior—even in the limit. We show how the relaxation scheme can be used in combination with a standard interior-point method to achieve superlinear convergence. Numerical results on the MacMPEC test problem set demonstrate the fast local convergence properties of the approach. Key words. nonlinear programming, mathematical programs with equilibrium constraints, complementarity constraints, constrained minimization, interior-point methods, primal-dual methods,

This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential quadratic programming (SQP) and sequential linear-quadratic programming (SLQP) methods that use trust regions to promote convergence. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.

This paper reviews the development of exact penalty methods for nonlinear optimization and discusses their increasingly important role in optimization algorithms and software. In their most recent stage of development, penalty methods adjust the penalty parameter dynamically. By controlling the degree of linear feasibility achieved at every iteration, these methods balance progress toward optimality and feasibility. The choice of the penalty parameter thus ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential linear-quadratic penalty methods, and is then extended to sequential quadratic programming. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.

Abstract—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 operator, while the leaders are the dominant suppliers. The ac approach allows one to capture the strategic behavior of suppliers regarding not only active but also reactive power. In addition, the impact of voltage and apparent power flow constraints can be analyzed. Different case studies are presented using a three-node system to highlight the features of the formulation. Results on a 14-node system are also presented. Index Terms—Competition, complementarity, Cournot, mathematical problem with equilibrium constraints (MPEC), Nash equilibrium, oligopoly. I.

Abstract—Deregulated electricity markets in the U.S. currently use an auction mechanism that minimizes total supply bid costs to select bids and their levels. Payments are then settled based on market-clearingprices. Under this setup, the consumer payments could be significantly higher than the minimized bid costs obtained from auctions. This gives rise to “payment cost minimization,” an alternative auction mechanism that minimizes consumer payments. We previously presented an augmented Lagrangian and surrogate optimization framework to solve payment cost minimization problems without considering transmission. This paper extends that approach to incorporate transmission capacity constraints. The consideration of transmission constraints complicates the problem by entailing power flow and introducing locational marginal orices (LMPs). DC power flow is used for simplicity and LMPs are defined by “economic dispatch ” for the selected supply bids. To characterize LMPs that appear in the payment cost objective function, Karush–Kuhn–Tucker (KKT) conditions of economic dispatch are established and embedded as constraints. The reformulated problem is difficult in view of the complex role of LMPs and the violation of constraint qualifications caused by the complementarity constraints of KKT conditions. Our key idea is to extend the surrogate optimization framework and use a regularization technique. Specific methods to satisfy the “surrogate optimization condition ” in the presence of transmission capacity constraints are highlighted. Numerical testing results of small examples and the IEEE Reliability Test System with randomly generated supply bids demonstrate the quality, effectiveness, and scalability of the method. Index Terms—Deregulated electricity markets, electricity auctions, locational marginal price (LMP), mathematical programs with equilibrium constraints, payment cost minimization, surrogate optimization, transmission constraints. I.

We propose a new approach to convex nonlinear multiobjective optimization that captures the geometry of the Pareto set by generating a discrete set of Pareto points optimally. We show that the problem of finding an optimal representation of the Pareto surface can be formulated as a mathematical program with complementarity constraints. The complementarity constraints arise from modeling the set of Pareto points, and the objective maximizes some quality measure of this discrete set. We present encouraging numerical experience on a range of test problem collected from the literature.

Abstract—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 modeled as a leader, while the central market operator running a cost minimization process is the sole follower. An ac model is considered to represent the transmission system. A three-node system is used to illustrate several cases, and study the implications of the incentives of the strategic players to exploit active and reactive power, and spinning reserves in order to maximize profits. Results for a 14-node system are also presented. Index Terms—Ac system, complementarity, market power, mathematical problem with complementarity constraints,

Line search algorithms for nonlinear programming must include safeguards to enjoy global convergence properties. This paper describes an exact penalization approach that extends the class of problems that can be solved with line search SQP methods. In the new algorithm, the penalty parameter is adjusted at every iteration to ensure sufficient progress in linear feasibility and to promote acceptance of the step. A trust region is used to assist in the determination of the penalty parameter (but not in the step computation). It is shown that the algorithm enjoys favorable global convergence properties. Numerical experiments illustrate the behavior of the algorithm on various difficult situations. 1