A smoothing-regularization approach to mathematical programs with vanishing constraints

We consider a numerical approach for the solution of a difficult class of optimization problems called mathematical programs with vanishing constraints. The basic idea is to reformulate the characteristic constraints of the program via a nonsmooth function and to eventually smooth it and regularize the feasible set with the aid of a certain smoothing- and regularization parameter t>0 such that the resulting problem is more tractable and coincides with the original program for t=0. We investigate the convergence behavior of a sequence of stationary points of the smooth and regularized problems by letting t tend to zero. Numerical results illustrating the performance of the approach are given. In particular, a large-scale example from topology optimization of mechanical structures with local stress constraints is investigated.     

A penalized Fischer-Burmeister NCP-function

We introduce a new NCP-function in order to reformulate the nonlinear complementarity problem as a nonsmooth system of equations. This new NCP-function turns out to have stronger theoretical properties than the widely used Fischer-Burmeister function and other NCP-functions suggested previously. Moreover, numerical experience indicates that a semismooth Newton method based on this new NCP-function performs considerably better than the corresponding method based on the Fischer-Burmeister function. Received: March 10, 1997 / Accepted: February 15, 2000?Published online May 12, 2000     

A QP-free constrained Newton-type method for variational inequality problems

Received January 5, 1997 / Revised version received November 19, 1997 Published online November 24, 1998     

Equivalence of the generalized complementarity problem to differentiable unconstrained minimization

We consider an unconstrained minimization reformulation of the generalized complementarity problem (GCP). The merit function introduced here is differentiable and has the property that its global minimizers coincide with the solutions of GCP. Conditions for its stationary points to be global minimizers are given. Moreover, it is shown that the level sets of the merit function are bounded under suitable assumptions. We also show that the merit function provides global error bounds for GCP. These results are based on a condition which reduces to the condition of the uniform P-function when GCP is specialized to the nonlinear complementarity problem. This condition also turns out to be useful in proving the existence and uniqueness of a solution for GCP itself. Finally, we obtain as a byproduct an error bound result with the natural residual for GCP.We thank Jong-Shi Pang for his valuable comments on error bound results with the natural residual for the nonlinear complementarity problem. We are also grateful to the anonymous referees for some helpful comments. The research of the second author was supported in part by the Science Research Grant-in-Aid from the Ministry of Education, Science, and Culture, Japan.     

Equivalence of Two Nondegeneracy Conditions for Semidefinite Programs

Nondegeneracy assumptions are often needed in order to prove the local fast convergence of suitable algorithms as well as in the sensitivity analysis for semidefinite programs. One of the more standard nondegeneracy conditions is the geometric condition used by Alizadeh et al. (Math. Program. 77:111–128, 1997). On the other hand, Kanzow and Nagel (SIAM J. Optim. 15:654–672, 2005) recently introduced an algebraic condition that was used in order to prove, for the first time, the local quadratic convergence of a suitable algorithm for the solution of semidefinite programs without using the strict complementarity assumption. The aim of this paper is to show that these two nondegeneracy conditions are equivalent.     

An interior-point affine-scaling trust-region method for semismooth equations with box constraints

An algorithm for the solution of a semismooth system of equations with box constraints is described. The method is an affine-scaling trust-region method. All iterates generated by this method are strictly feasible. In this way, possible domain violations outside or on the boundary of the box are avoided. The method is shown to have strong global and local convergence properties under suitable assumptions, in particular, when the method is used with a special scaling matrix. Numerical results are presented for a number of problems arising from different areas.     

An augmented Lagrangian method for optimization problems with structured geometric constraints

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints are nonconvex and possibly complicated, but allow for a fast computation of projections onto this nonconvex set. Typical problem classes which satisfy this requirement are optimization problems with disjunctive constraints (like complementarity or cardinality constraints) as well as optimization problems over sets of matrices which have to satisfy additional rank constraints. The key idea behind our method is to keep these complicated constraints explicitly in the constraints and to penalize only the remaining constraints by an augmented Lagrangian function. The resulting subproblems are then solved with the aid of a problem-tailored nonmonotone projected gradient method. The corresponding convergence theory allows for an inexact solution of these subproblems. Nevertheless, the overall algorithm computes so-called Mordukhovich-stationary points of the original problem under a mild asymptotic regularity condition, which is generally weaker than most of the respective available problem-tailored constraint qualifications. Extensive numerical experiments addressing complementarity- and cardinality-constrained optimization problems as well as a semidefinite reformulation of MAXCUT problems visualize the power of our approach.

     

Nonlinear complementarity as unconstrained optimization

Several methods for solving the nonlinear complementarity problem (NCP) are developed. These methods are generalizations of the recently proposed algorithms of Mangasarian and Solodov (Ref. 1) and are based on an unconstrianed minimization formulation of the nonlinear complementarity problem. It is shown that, under certain assumptions, any stationary point of the unconstrained objective function is already a solution of NCP. In particulr, these assumptions are satisfied by the mangasarian and Soolodov implicit Lagranian functioin. Furthermore, a special Newton-type method is suggested, and conditions for its local quadratic convergence are given. Finally, some preliminary numerical results are presented.The author would like to thank Dr. Oswald Knoth (Leipzig) for pointing out that the equivalence of Lemma 2.2. is not true for complementarity problems which have no solutions. He is also grateful to the anonymous referencees for their helpful comments.     

On the resolution of monotone complementarity problems

A reformulation of the nonlinear complementarity problem (NCP) as an unconstrained minimization problem is considered. It is shown that any stationary point of the unconstrained objective function is a solution of NCP if the mapping F involved in NCP is continuously differentiable and monotone, and that the level sets are bounded if F is continuous and strongly monotone. A descent algorithm is described which uses only function values of F. Some numerical results are given.     

Optimality Conditions for Disjunctive Programs with Application to Mathematical Programs with Equilibrium Constraints

We consider optimization problems with a disjunctive structure of the feasible set. Using Guignard-type constraint qualifications for these optimization problems and exploiting some results for the limiting normal cone by Mordukhovich, we derive different optimality conditions. Furthermore, we specialize these results to mathematical programs with equilibrium constraints. In particular, we show that a new constraint qualification, weaker than any other constraint qualification used in the literature, is enough in order to show that a local minimum results in a so-called M-stationary point. Additional assumptions are also discussed which guarantee that such an M-stationary point is in fact a strongly stationary point.