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Interiorpoint methods for nonconvex nonlinear programming: Filter methods and merit functions
 Computational Optimization and Applications
, 2002
"... Abstract. In this paper, we present global and local convergence results for an interiorpoint 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 ..."
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Cited by 84 (7 self)
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Abstract. In this paper, we present global and local convergence results for an interiorpoint 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 interiorpoint method [12]. The global convergence of the algorithm is proved under mild assumptions, and local analysis shows that it converges Qquadratically 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 KarushKuhnTucker 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 highlyefficient nonlinear solver and thoroughly analyze the effects of enforcing theoretical convergence guarantees on the computational performance of the algorithm. 1.
Some properties of regularization and penalization schemes for MPECs
 Optimization Methods and Software
, 2004
"... Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorde ..."
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Cited by 19 (2 self)
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Abstract. Some properties of regularized and penalized nonlinear programming formulations of mathematical programs with equilibrium constraints (MPECs) are described. The focus is on the properties of these formulations near a local solution of the MPEC at which strong stationarity and a secondorder sufficient condition are satisfied. In the regularized formulations, the complementarity condition is replaced by a constraint involving a positive parameter that can be decreased to zero. In the penalized formulation, the complementarity constraint appears as a penalty term in the objective. Existence and uniqueness of solutions for these formulations are investigated, and estimates are obtained for the distance of these solutions to the MPEC solution under various assumptions.
Elasticmode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties
 Math. Program
, 2005
"... Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the ..."
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Cited by 11 (1 self)
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Abstract. The elasticmode formulation of the problem of minimizing a nonlinear function subject to equilibrium constraints has appealing local properties in that, for a finite value of the penalty parameter, local solutions satisfying first and secondorder necessary optimality conditions for the original problem are also first and secondorder points of the elasticmode formulation. Here we study global convergence properties of methods based on this formulation, which involve generating an (exact or inexact) first or secondorder point of the formulation, for nondecreasing values of the penalty parameter. Under certain regularity conditions on the active constraints, we establish finite or asymptotic convergence to points having a certain stationarity property (such as strong stationarity, Mstationarity, or Cstationarity). Numerical experience with these approaches is discussed. In particular, our analysis and the numerical evidence show that exact complementarity can be achieved finitely even when the elasticmode formulation is solved inexactly. Key words. Nonlinear programming, equilibrium constraints, complementarity constraints, elasticmode formulation, strong stationarity, Cstationarity, Mstationarity. AMS subject classifications 49M30, 49M37, 65K05, 90C30, 90C33 1.
Semismooth Newton method for the lifted reformulation of mathematical programs with complementarity constraints
, 2009
"... We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismo ..."
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Cited by 5 (4 self)
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We consider a reformulation of mathematical programs with complementarity constraints, where by introducing an artificial variable the constraints are converted into equalities which are once but not twice differentiable. We show that the Lagrange optimality system of such a reformulation is semismooth and BDregular at the solution under reasonable assumptions. Thus, fast local convergence can be obtained by applying the semismooth Newton method. Moreover, it turns out that the squared residual of the Lagrange system is continuously differentiable (even though the system itself is not), which opens the way for a natural globalization of the local algorithm.
A TrustRegion Method for Nonlinear Bilevel Programming: Algorithm and Computational Experience
, 2005
"... We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective ..."
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Cited by 4 (0 self)
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We consider the approximation of nonlinear bilevel mathematical programs by solvable programs of the same type, i.e., bilevel programs involving linear approximations of the upperlevel objective and all constraintdefining functions, as well as a quadratic approximation of the lowerlevel objective. We describe the main features of the algorithm and the resulting software. Numerical experiments tend to confirm the promising behavior of the method.
SEMISMOOTH SQP METHOD FOR EQUALITYCONSTRAINED OPTIMIZATION PROBLEMS WITH AN APPLICATION TO THE LIFTED REFORMULATION OF MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS
, 2010
"... We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our ..."
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Cited by 2 (2 self)
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We consider the sequential quadratic programming algorithm (SQP) applied to equalityconstrained optimization problems, where the problem data is differentiable with Lipschitzcontinuous first derivatives. For this setting, DennisMoré type analysis of primal superlinear convergence is presented. Our main motivation is a special modification of SQP tailored to the structure of the lifted reformulation of mathematical programs with complementarity constraints (MPCC). For this problem, we propose a special positive definite modification of the matrices in the generalized Hessian, which is suitable for globalization of SQP based on the penalty function, and at the same time can be expected to satisfy our general DennisMoré type conditions, thus preserving local superlinear convergence. (Standard quasiNewton updates in the SQP framework require twice differentiability of the problem data at the solution for superlinear convergence.) Preliminary numerical results comparing a number of quasiNewton versions of semismooth SQP applied to MPCC are also reported. Key words: sequential quadratic programming, semismoothness, Bdifferential, BDregularity, semismooth Newton method, secondorder sufficiency, mathematical programs with complementarity constraints.
PROGRAMMING PROBLEM
"... Abstract. In this paper, we investigate the application of feasible direction method for an optimistic nonlinear bilevel programming problem. The convex lower level problem of an optimistic nonlinear bilevel programming problem is replaced by relaxed KKT conditions. The feasible direction method dev ..."
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Abstract. In this paper, we investigate the application of feasible direction method for an optimistic nonlinear bilevel programming problem. The convex lower level problem of an optimistic nonlinear bilevel programming problem is replaced by relaxed KKT conditions. The feasible direction method developed by Topkis and Veinott [22] is applied to the auxiliary problem to get a Bouligand stationary point for an optimistic bilevel programming problem. (1.1) 1.
Preprint ANL/MCSP11430404 GLOBAL CONVERGENCE OF AN ELASTIC MODE APPROACH FOR A CLASS OF MATHEMATICAL PROGRAMS WITH COMPLEMENTARITY CONSTRAINTS
"... Abstract. We prove that any accumulation point of an elastic mode approach, applied to the optimization of a mixed P variational inequality, that approximately solves the relaxed subproblems is a Cstationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addi ..."
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Abstract. We prove that any accumulation point of an elastic mode approach, applied to the optimization of a mixed P variational inequality, that approximately solves the relaxed subproblems is a Cstationary point of the problem of optimizing a parametric mixed P variational inequality. If, in addition, the accumulation point satisfies the MPCCLICQ constraint qualification and if the solutions of the subproblem satisfy approximate secondorder sufficient conditions, then the limiting point is an Mstationary point. Moreover, if the accumulation point satisfies the upperlevel strict complementarity condition, the accumulation point will be a strongly stationary point. If we assume that the penalty function associated with the feasible set of the mathematical program with complementarity constraints has bounded level sets and if the objective function is bounded below, we show that the algorithm will produce bounded iterates and will therefore have at least one accumulation point. We prove that the obstacle problem satisfies our assumptions for both a rigid and a deformable obstacle. The theoretical conclusions are validated by several numerical examples. Key words. MPCC, global convergence, complementarity constraints, nonlinear programming, obstacle problem. AMS subject classifications. 65K10, 90C33.