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Abstract. We consider equality-constrained optimization problems, where a given solution may not satisfy any constraint qualification but satisfies the standard second-order sufficient condition for optimality. Based on local identification of the rank of the constraints degeneracy via the singular-value decomposition, we derive a modified primal-dual optimality system whose solution is locally unique, nondegenerate, and thus can be found by standard Newton-type techniques. Using identification of active constraints, we further extend our approach to mixed equality-and inequality-constrained problems, and to mathematical programs with complementarity constraints (MPCC). In particular, for MPCC we obtain a local algorithm with quadratic convergence under the second-order sufficient condition only, without any constraint qualifications, not even the special MPCC constraint qualifications.

We propose and analyze a perturbed version of the classical JosephyNewton method for solving generalized equations. This perturbed framework is convenient to treat in a unified way standard sequential quadratic programming, its stabilized version, sequential quadratically constrained quadratic programming, and linearly constrained Lagrangian methods. For the linearly constrained Lagrangian methods, in particular, we obtain superlinear convergence under the second-order sufficient optimality condition and the strict Mangasarian-Fromovitz constraint qualification, while previous results in the literature assume (in addition to second-order sufficiency) the stronger linear independence constraint qualification as well as the strict complementarity condition. For the sequential quadratically constrained quadratic programming methods, we prove primal-dual superlinear/quadratic convergence under the same assumptions as above, which also gives a new result.

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We obtain local estimates of the distance to a set defined by equality constraints under assumptions which are weaker than those previously used in the literature. Specifically, we assume that the constraints mapping has a Lipschitzian derivative, and satisfies a certain 2-regularity condition at the point under consideration. This setting directly subsumes the classical regular case and the twice differentiable 2-regular case, for which error bounds are known, but it is significantly richer than either of these two cases. When applied to a certain equation-based reformulation of the nonlinear complementarity problem, our results yield an error bound under an assumption more general than b-regularity. The latter appears to be the weakest assumption under which a local error bound for complementarity problems was previously available. We also discuss an application of our results to the convergence rate analysis of the exterior penalty method for solving irregular problems.Key words. error bound -C 1,1 -mapping -2-regularity -nonlinear complementarity problem -exterior penalty -rate of convergence Error bounds and their applicationsAmong the most important tools for theoretical and numerical treatment of nonlinear operator equations, optimization problems, variational inequalities, and other related problems, are the so-called error bounds, i.e., upper estimates of the distance to a given set in terms of some residual function. We refer the reader to [31] for a survey of error bounds and their applications. When the set is defined by functional constraints, a typical residual function is some measure of violation of constraints at the given point. When available, error bounds can often be used to obtain a constructive local description of the set under consideration. This description, in turn, plays a central role in the theory of optimality conditions. As another important application of error bounds, we mention development, implementation, and convergence rate analysis of numerical methods for solving optimization and related problems.In this paper, we consider error bounds for sets given by equality constraints. However, it is worth to point out that this setting implicitly includes sets of more general structure, namely those whose constraints can be (equivalently) reformulated into equations. One such example is solution set of the nonlinear complementarity problem, which we shall study in Sect. 3.

The stabilized version of the sequential quadratic programming algorithm (sSQP) had been developed in order to achieve superlinear convergence in situations when the Lagrange multipliers associated to a solution are not unique. Within the framework of Fischer (Math Program 94:91-124, 2002), the key to local superlinear convergence of sSQP are the following two properties: upper Lipschitzian behavior of solutions of the Karush-Kuhn-Tucker (KKT) system under canonical perturbations and local solvability of sSQP subproblems with the associated primal-dual step being of the order of the distance from the current iterate to the solution set of the unperturbed KKT system. According to Fernández and Solodov (Math Program 125:47-73, 2010), both of these properties are ensured by the second-order sufficient optimality condition (SOSC) without any constraint qualification assumptions. In this paper, we state precise relationships between the upper Lipschitzian property of solutions of KKT systems, error bounds for KKT systems, the notion of critical Lagrange multipliers (a subclass of multipliers that violate SOSC in a very special way), the second-order necessary condition for optimality, and solvability of sSQP subproblems. Moreover, for the problem with equality constraints only, we prove superlinear convergence of sSQP under the assumption that the dual starting point is close to a noncritical multiplier. Since noncritical multipliers include all those satisfying SOSC but are not limited to them, we believe this gives the first superlinear convergence result for any Newtonian method for constrained optimization under assumptions that do not include any constraint qualifications and are weaker than SOSC. In the general case when inequality constraints are present, we show that such a relaxation of assumptions is not possible. We also consider applying sSQP to the problem where inequality constraints are reformulated into equalities using slack variables, and discuss the assumptions needed for convergence in this approach. We conclude with consequences for local regularization methods proposed in (Izmailov and Solodov SIAM J Optim 16:210-228, 2004; Wright SIAM J. Optim. 15:673-676, 2005). In particular, we show that these methods are still locally superlinearly convergent under the noncritical multiplier assumption, weaker than SOSC employed originally.

Abstract. We consider global convergence properties of the augmented Lagrangian methods on problems with degenerate constraints, with a special emphasis on mathematical programs with complementarity constraints (MPCC). In the general case, we show convergence to stationary points of the problem under an error bound condition for the feasible set (which is weaker than constraint qualifications), assuming that the iterates have some modest features of approximate local minimizers of the augmented Lagrangian. For MPCC, we first argue that even weak forms of general constraint qualifications that are suitable for convergence of the augmented Lagrangian methods, such as the recently proposed relaxed positive linear dependence condition, should not be expected to hold and thus special analysis is needed. We next obtain a rather complete picture, showing that, under this context's usual MPCC-linear independence constraint qualification, feasible accumulation points of the iterates are guaranteed to be C-stationary for MPCC (better than weakly stationary), but in general need not be M-stationary (hence, neither strongly stationary). However, strong stationarity is guaranteed if the generated dual sequence is bounded, which we show to be the typical numerical behavior even though the multiplier set itself is unbounded. Experiments with the ALGENCAN augmented Lagrangian solver on the MacMPEC and DEGEN collections are reported, with comparisons to the SNOPT and filterSQP implementations of the SQP method, to the MINOS implementation of the linearly constrained Lagrangian method, and to the interior-point solvers IPOPT and KNITRO. We show that ALGENCAN is a very good option if one is primarily interested in robustness and quality of computed solutions.

We study local structure of a nonlinear mapping near points where standard regularity and/or smoothness assumptions need not be satisfied. We introduce a new concept of 2-regularity (a certain kind of second-order regularity) for a once differentiable mapping whose derivative is Lipschitz continuous. Under this 2-regularity condition, we obtain the representation theorem and the covering theorem (i.e., stability with respect to "right-hand side" perturbations) under assumptions that are weaker than those previously employed in the literature for results of this type. These results are further used to derive a constructive description of the tangent cone to a set defined by (2-regular) equality constraints and optimality conditions for related optimization problems. The class of mappings introduced and studied in the paper appears to be a convenient tool for treating complementarity structures by means of an appropriate equation-based reformulation. Optimality conditions for mathematical programs with (equivalently reformulated) complementarity constraints are also discussed.1. Introduction. In this paper, we consider the problem of local approximation of a nonlinear mapping near a given point in the absence of standard combinations of smoothness and regularity assumptions. We further study consequences of this approximation, in particular those relevant for constructive description of tangent directions to level surfaces, and optimality conditions for problems with feasible regions defined by irregular mappings.Let X and Y be Banach spaces, and consider a pointx ∈ X, its neighborhood V in X, and a mapping F V → Y . In what follows, X Y stands for the space of continuous linear operators from X to Y . For a linear operator ∈ X Y , ker = x ∈ X x = 0 is its null space, im = y ∈ Y y = x for some x ∈ X is its image space, and * Y * → X * is the linear operator adjoint to , where X * and Y * denote the dual spaces of X and Y , respectively.It is well known that if F is (first-order) regular atx, i.e.,