For constrained equations with nonisolated solutions, we show that if the equation mapping is 2-regular at a given solution with respect to a direction in the null space of the Jacobian, and this direction is interior feasible, then there is an associated domain of starting points from which a family of Newton-type methods is well-de ned … Read more
We consider fully convex lower level standard optimistic bilevel problems. We show that this class of mathematical programs is sufficiently broad to encompass significant real-world applications. We establish that the critical points of a relaxation of the original problem correspond to the equilibria of a suitably defined generalized Nash equilibrium problem. The latter game is … Read more
We consider nonlinear and nonsmooth mixing aspects in gas transport optimization problems. As mixed-integer reformulations of pooling-type mixing models already render small-size instances computationally intractable, we investigate the applicability of smooth nonlinear programming techniques for equivalent complementarity-based reformulations. Based on recent results for remodeling piecewise affine constraints using an inverse parametric quadratic programming approach, we … Read more
We introduce two new weaker Constraint Qualifications (CQs) for Mathematical Programs with Equilibrium (or Complementarity) Constraints, MPEC for short. One of them is a tailored version of the Constant Rank of Subspace Component (CRSC) and the other is a relaxed version of the MPEC-No Nonzero Abnormal Multiplier Constraint Qualification (MPEC-NNAMCQ). Both incorporate the exact set … Read more
We propose a new family of relaxation schemes for mathematical program with complementarity constraints that extends the relaxations of Kadrani, Dussault, Bechakroun from 2009 and the one of Kanzow \& Schwartz from 2011. We discuss the properties of the sequence of relaxed non-linear program as well as stationarity properties of limiting points. A sub-family of … Read more
We discuss here the convergence of relaxation methods for MPCC with approximate sequence of stationary points by presenting a general framework to study these methods. It has been pointed out in the literature, \cite{kanzow2015}, that relaxation methods with approximate stationary points fail to give guarantee of convergence. We show that by defining a new strong … Read more
By reviewing the primal-dual hybrid algorithm (PDHA) proposed by He, You and Yuan (SIAM J. Imaging Sci. 2014;7(4):2526-2537), in this paper we introduce four improved schemes for solving a class of generalized saddle-point problems. By making use of the variational inequality, weaker conditions are presented to ensure the global convergence of the proposed algorithms, where … Read more
Given a closed convex cone C in a finite dimensional real Hilbert space H, a weakly homogeneous map f:C–>H is a sum of two continuous maps h and g, where h is positively homogeneous of (positive) degree gamma on C and g(x)/||x||^gamma–>0 as ||x||–>infinity in C. Given such a map f, a nonempty closed convex … Read more
Mathematical Programs with Complementarity Constraints (MPCCs) are difficult optimization problems that do not satisfy the majority of the usual constraint qualifications (CQs) for standard nonlinear optimization. Despite this fact, classical methods behaves well when applied to MPCCs. Recently, Izmailov, Solodov and Uskov proved that first order augmented Lagrangian methods, under a natural adaption of the … Read more
The proximal ADMM which adds proximal regularizations to ADMM’s subproblems is a popular and useful method for linearly constrained separable convex problems, especially its linearized case. A well-known requirement on guaranteeing the convergence of the method in the literature is that the proximal regularization must be positive semidefinite. Recently it was shown by He et … Read more