Joaquim J. Júdice

Descent approaches for quadratic bilevel programming

The bilevel programming problem involves two optimization problems where the data of the first one is implicitly determined by the solution of the second. In this paper, we introduce two descent methods for a special instance of bilevel programs where the inner problem is strictly convex quadratic. The first algorithm is based on pivot steps and may not guarantee local optimality. A modified steepest descent algorithm is presented to overcome this drawback. New rules for computing exact stepsizes are introduced and a hybrid approach that combines both strategies is discussed. It is proved that checking local optimality in bilevel programming is a NP-hard problem.

Discrete linear bilevel programming problem

In this paper, we analyze some properties of the discrete linear bilevel program for different discretizations of the set of variables. We study the geometry of the feasible set and discuss the existence of an optimal solution. We also establish equivalences between different classes of discrete linear bilevel programs and particular linear multilevel programming problems. These equivalences are based on concave penalty functions and can be used to design penalty function methods for the solution of discrete linear bilevel programs.

A sequential LCP method for bilevel linear programming

In this paper, we discuss an SLCP algorithm for the solution of Bilevel Linear Programs (BLP) which consists of solving a sequence of Linear Complementarity Problems (LCP) by using a hybrid enumerative method. This latter algorithm incorporates a number of procedures that reduce substantially the search for a solution of the LCP or for showing that the LCP has no solution. Computational experience with the SLCP algorithm shows that it performs quite well for the solution of small- and medium-scale BLPs with sparse structure. Furthermore, the algorithm is shown to be more efficient than a branch-and-bound method for solving the same problems.

An Active Set Newton Algorithm for Large-Scale Nonlinear Programs with Box Constraints

A new algorithm for large-scale nonlinear programs with box constraints is introduced. The algorithm is based on an efficient identification technique of the active set at the solution and on a nonmonotone stabilization technique. It possesses global and superlinear convergence properties under standard assumptions. A new technique for generating test problems with known characteristics is also introduced. The implementation of the method is described along with computational results for large-scale problems.

A comparison of block pivoting and interior-point algorithms for linear least squares problems with nonnegative variables

In this paper we discuss the use of block principal pivoting and predictor-corrector methods for the solution of large-scale linear least squares problems with nonnegative variables (NVLSQ). We also describe two implementations of these algorithms that are based on the normal equations and corrected seminormal equations (CSNE) approaches. We show that the method of normal equations should be employed in the implementation of the predictor-corrector algorithm. This type of approach should also be used in the implementation of the block principal pivoting method, but a switch to the CSNE method may be useful in the last iterations of the algorithm. Computational experience is also included in this paper and shows that both the predictor-corrector and the block principal pivoting algorithms are quite efficient to deal with large-scale NVLSQ problems.

The eigenvalue complementarity problem

Abstract In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.

The symmetric eigenvalue complementarity problem

In this paper the Eigenvalue Complementarity Problem (EiCP) with real symmetric matrices is addressed. It is shown that the symmetric (EiCP) is equivalent to finding an equilibrium solution of a differentiable optimization problem in a compact set. A necessary and sufficient condition for solvability is obtained which, when verified, gives a convenient starting point for any gradient-ascent local optimization method to converge to a solution of the (EiCP). It is further shown that similar results apply to the Symmetric Generalized Eigenvalue Complementarity Problem (GEiCP). Computational tests show that these reformulations improve the speed and robustness of the solution methods.

A truncated primal-infeasible dual-feasible network interior point method

In this paper, we introduce the truncated primal-infeasible dual-feasible interior point algorithm for linear programming and describe an implementation of this algorithm for solving the minimum-cost network flow problem. In each iteration, the linear system that determines the search direction is computed inexactly, and the norm of the resulting residual vector is used in the stopping criteria of the iterative solver employed for the solution of the system. In the implementation, a preconditioned conjugate gradient method is used as the iterative solver. The details of the implementation are described and the code PDNET is tested on a large set of standard minimum-cost network flow test problems. Computational results indicate that the implementation is competitive with state-of-the-art network flow codes.

On the asymmetric eigenvalue complementarity problem

In this paper, we discuss the eigenvalue complementarity problem (EiCP) where at least one of its defining matrices is asymmetric. A sufficient condition for the existence of a solution to the EiCP is established. The EiCP is shown to be equivalent to finding a global minimum of an appropriate merit function on a convex set Ω defined by linear constraints. A sufficient condition for a stationary point of this function on Ω to be a solution of the EiCP is presented. A branch-and-bound procedure is developed for finding a global minimum of this merit function on Ω. In addition, a sequential enumerative algorithm for the computation of the minimum and the maximum eigenvalues is also discussed. Computational experience is included to highlight the efficiency and efficacy of the proposed methodologies to solve the asymmetric EiCP.

Algorithms for Linear Complementarity Problems

This paper presents a survey of the Linear Complementarity Problem (LCP). The most important existence and complexity results of the LCP are first reviewed. Direct, iterative and enumerative algorithms are then discussed together with their benefits and drawbacks.