Calmness of the Feasible Set Mapping for Linear Inequality Systems

In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.     

Strong calmness of perturbed KKT system for a class of conic programming with degenerate solutions

ABSTRACT

This paper is concerned with the strong calmness of the KKT solution mapping for a class of canonically perturbed conic programming, which plays a central role in achieving fast convergence under situations when the Lagrange multiplier associated to a solution of these conic optimization problems is not unique. We show that the strong calmness of the KKT solution mapping is equivalent to a local error bound for the solutions to the perturbed KKT system, and is also equivalent to the pseudo-isolated calmness of the stationary point mapping along with the calmness of the multiplier set mapping at the corresponding reference point. Sufficient conditions are also provided for the strong calmness by establishing the pseudo-isolated calmness of the stationary point mapping in terms of the noncriticality of the associated multiplier, and the calmness of the multiplier set mapping in terms of a relative interior condition for the multiplier set. These results cover and extend the existing ones in Hager and Gowda [Stability in the presence of degeneracy and error estimation. Math Program. 1999;85:181–192]; Izmailov and Solodov [Stabilized SQP revisited. Math Program. 2012;133:93–120] for nonlinear programming and in Cui et al. [On the asymptotic superlinear convergence of the augmented Lagrangian method for semidefinite programming with multiple solutions. 2016, arXiv: 1610.00875v1]; Zhang and Zhang [Critical multipliers in semidefinite programming. 2018, arXiv: 1801.02218v1] for semidefinite programming.     

Critical Objective Size and Calmness Modulus in Linear Programming

This paper introduces the concept of critical objective size associated with a linear program in order to provide operative point-based formulas (only involving the nominal data, and not data in a neighborhood) for computing or estimating the calmness modulus of the optimal set (argmin) mapping under uniqueness of nominal optimal solution and perturbations of all coefficients. Our starting point is an upper bound on this modulus given in Cánovas et al. (4). In this paper we prove that this upper bound is attained if and only if the norm of the objective function coefficient vector is less than or equal to the critical objective size. This concept also allows us to obtain operative lower bounds on the calmness modulus. We analyze in detail an illustrative example in order to explore some strategies that can improve the referred upper and lower bounds.     

A New Complexity Analysis for Full-Newton Step Infeasible Interior-Point Algorithm for Horizontal Linear Complementarity Problems

In this paper, we first present a full-Newton step feasible interior-point algorithm for solving horizontal linear complementarity problems. We prove that the full-Newton step to the central path is quadratically convergent. Then, we generalize an infeasible interior-point method for linear optimization to horizontal linear complementarity problems based on new search directions. This algorithm starts from strictly feasible iterates on the central path of a perturbed problem that is produced by a suitable perturbation in the horizontal linear complementarity problem. We use the so-called feasibility steps that find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain a strictly feasible iterate close enough to the central path of the new perturbed problem. The complexity of the algorithm coincides with the best known iteration bound for infeasible interior-point methods.     

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.     

Stability for differential mixed variational inequalities

In this paper, an existence theorem of Carathéodory weak solution for a differential mixed variational inequality is presented under suitable conditions. Furthermore, some upper semicontinuity and continuity results concerned with the Carathéodory weak solution set mapping for the differential mixed variational inequality are given when both the mapping and the constraint set are perturbed by two different parameters.     

Stability Analysis for Minty Mixed Variational Inequality in Reflexive Banach Spaces

This paper is devoted to the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. Several equivalent characterizations are given for the Minty mixed variational inequality to have nonempty and bounded solution set. A stability result is presented for the Minty mixed variational inequality with Φ-pseudomonotone mapping in reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. As an application, a stability result for a generalized mixed variational inequality is also obtained. The results presented in this paper generalize and extend some known results in Fan and Zhong (Nonlinear Anal., Theory Methods Appl. 69:2566–2574, 2008) and He (J. Math. Anal. Appl. 330:352–363, 2007).     

Multiobjective optimization problem with variational inequality constraints

 We study a general multiobjective optimization problem with variational inequality, equality, inequality and abstract constraints. Fritz John type necessary optimality conditions involving Mordukhovich coderivatives are derived. They lead to Kuhn-Tucker type necessary optimality conditions under additional constraint qualifications including the calmness condition, the error bound constraint qualification, the no nonzero abnormal multiplier constraint qualification, the generalized Mangasarian-Fromovitz constraint qualification, the strong regularity constraint qualification and the linear constraint qualification. We then apply these results to the multiobjective optimization problem with complementarity constraints and the multiobjective bilevel programming problem. Received: November 2000 / Accepted: October 2001 Published online: December 19, 2002 Key Words. Multiobjective optimization – Variational inequality – Complementarity constraint – Constraint qualification – Bilevel programming problem – Preference – Utility function – Subdifferential calculus – Variational principle Research of this paper was supported by NSERC and a University of Victoria Internal Research Grant Research was supported by the National Science Foundation under grants DMS-9704203 and DMS-0102496 Mathematics Subject Classification (2000): Sub49K24, 90C29     

Hölder Error Bounds and Hölder Calmness with Applications to Convex Semi-infinite Optimization

Using techniques of variational analysis, necessary and sufficient subdifferential conditions for Hölder error bounds are investigated and some new estimates for the corresponding modulus are obtained. As an application, we consider the setting of convex semi-infinite optimization and give a characterization of the Hölder calmness of the argmin mapping in terms of the level set mapping (with respect to the objective function) and a special supremum function. We also estimate the Hölder calmness modulus of the argmin mapping in the framework of linear programming.

     

Exterior Minimum-Penalty Path-Following Methods in Semidefinite Programming

A semidefinite programming problem is a mathematical program in which the objective function is linear in the unknowns and the constraint set is defined by a linear matrix inequality. This problem is nonlinear, nondifferentiable but convex. It covers several standard problems, such as linear and quadratic programming, and has many applications in engineering. In this paper, we introduce the notion of minimal-penalty path, which is defined as the collection of minimizers for a family of convex optimization problems, and propose two methods for solving the problem by following the minimal-penalty path from the exterior of the feasible set. Our first method, which is also a constraint-aggregation method, achieves the solution by solving a sequence of linear programs, but exhibits a zigzagging behavior around the minimal-penalty path. Our second method eliminates the above drawback by following efficiently the minimum-penalty path through the centering and ascending steps. The global convergence of the methods is proved and their performance is illustrated by means of an example.     

Metric regularity of semi-infinite constraint systems

We obtain a formula for the modulus of metric regularity of a mapping defined by a semi-infinite system of equalities and inequalities. Based on this formula, we prove a theorem of Eckart-Young type for such set-valued infinite-dimensional mappings: given a metrically regular mapping F of this kind, the infimum of the norm of a linear function g such that F+g is not metrically regular is equal to the reciprocal to the modulus of regularity of F. The Lyusternik-Graves theorem gives a straightforward extension of these results to nonlinear systems. We also discuss the distance to infeasibility for homogeneous semi-infinite linear inequality systems. Dedicated to R. T. Rockafellar on his 70th Birthday Research partially supported by grants BFM2002-04114-C02 (01-02) from MCYT (Spain) and FEDER (E.U.), GV04B-648 and GRUPOS04/79 from Generalitat Valenciana (Spain), and Bancaja-UMH (Spain).     

不等式约束随机非线性规划的扰动Karush-Kuhn-Tucker系统的解路径

论文聚焦概率测度发生扰动时的随机非线性规划的稳定性分析的研究.目标函数的Lipschitz连续性和可行集值映射的度量正则性条件可保证最优解集合的外半连续性和最优值的Lipschitz连续性.更重要地,本文证明了,如果原问题的极小点处线性无关约束规范和强二阶充分性条件成立,那么存在一Lipschitz连续的解路径满足扰动问题的Karush-Kuhn-Tucker条件.     

Strong Abadie CQ,ACQ, calmness and linear regularity

The Abadie CQ (ACQ) for convex inequality systems is a fundamental notion in optimization and approximation theory. In terms of the contingent cone and tangent derivative, we extend the Abadie CQ to more general convex multifunction cases and introduce the strong ACQ for both multifunctions and inequality systems. Some seemly unrelated notions are unified by the new ACQ and strong ACQ. Relationships among ACQ, strong ACQ, basic constraint qualification (BCQ) and strong BCQ are discussed. Using the strong ACQ, we study calmness of a closed and convex multifunction between two Banach spaces and, different from many existing dual conditions for calmness, establish several primal characterizations of calmness. As applications, some primal characterizations for error bounds and linear regularity are established; in particular, some existing results are improved.     

Enhanced Karush–Kuhn–Tucker condition and weaker constraint qualifications

In this paper we study necessary optimality conditions for nonsmooth optimization problems with equality, inequality and abstract set constraints. We derive the enhanced Fritz John condition which contains some new information even in the smooth case than the classical enhanced Fritz John condition. From this enhanced Fritz John condition we derive the enhanced Karush–Kuhn–Tucker condition and introduce the associated pseudonormality and quasinormality condition. We prove that either pseudonormality or quasinormality with regularity on the constraint functions and the set constraint implies the existence of a local error bound. Finally we give a tighter upper estimate for the Fréchet subdifferential and the limiting subdifferential of the value function in terms of quasinormal multipliers which is usually a smaller set than the set of classical normal multipliers. In particular we show that the value function of a perturbed problem is Lipschitz continuous under the perturbed quasinormality condition which is much weaker than the classical normality condition.     

Differential Vector Variational Inequalities in Finite-Dimensional Spaces

In this paper, a differential vector variational inequality is introduced and studied in finite-dimensional Euclidean spaces. The existence of a Carathéodory weak solution for the differential vector variational inequality is presented under some suitable conditions. Furthermore, the upper semicontinuity and the lower semicontinuity of the solution sets for the differential variational inequality are established when both the mapping and the constraint set are perturbed by two different parameters.     

Isolated calmness of solution mappings in convex semi-infinite optimization

This paper is concerned with isolated calmness of the solution mapping of a parameterized convex semi-infinite optimization problem subject to canonical perturbations. We provide a sufficient condition for isolated calmness of this mapping. This sufficient condition characterizes the strong uniqueness of minimizers, under the Slater constraint qualification. Moreover, on the assumption that the objective function and the constraints are linear, we show that this condition is also necessary for isolated calmness.     

Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

Mathematical Programming - This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in $${\mathbb {R}}^{n}$$ . To start with, we...     

Regularity modulus of arbitrarily perturbed linear inequality systems

We aim to quantify the stability of systems of (possibly infinitely many) linear inequalities under arbitrary perturbations of the data. Our focus is on the Aubin property (also called pseudo-Lipschitz) of the solution set mapping, or, equivalently, on the metric regularity of its inverse mapping. The main goal is to determine the regularity modulus of the latter mapping exclusively in terms of the system's data. In our context, both, the right- and the left-hand side of the system are subject to possible perturbations. This fact entails notable differences with respect to previous developments in the framework of linear systems with perturbations of the right-hand side. In these previous studies, the feasible set mapping is sublinear (which is not our current case) and the well-known Radius Theorem constitutes a useful tool for determining the modulus. In our current setting we do not have an explicit expression for the radius of metric regularity, and we have to tackle the modulus directly. As an application we approach, under appropriate assumptions, the regularity modulus for a semi-infinite system associated with the Lagrangian dual of an ordinary nonlinear programming problem.     

Uniqueness for Quasi-variational Inequalities

This paper presents a uniqueness result for a quasi-variational inequality QVI(1) that, in contrast to existing results, does not require the projection mapping on a variable closed and convex set to be a contraction. Our basic idea is to find a simple QVI(0), for example a variational inequality, for which we can show the existence of a unique solution. Further, exploiting some nonsingularity condition, we will guarantee the existence of a continuous solution path from the unique solution of QVI(0) to a solution of QVI(1). Finally, we can show that the existence of a second different solution of QVI(1) contradicts the nonsingularity condition. Moreover, we present some matrix-based sufficient conditions for our nonsingularity assumption, and we discuss these assumptions in the context of generalized Nash equilibrium problems with quadratic cost and affine linear constraint functions.     

A full-Newton step infeasible interior-point algorithm for monotone LCP based on a locally-kernel function

We propose a new full-Newton step infeasible interior-point algorithm for monotone linear complementarity problems based on a simple locally-kernel function. The algorithm uses the simple locally-kernel function to determine the search directions and define the neighborhood of central path. Two types of full-Newton steps are used, feasibility step and centering step. The algorithm starts from strictly feasible iterates of a perturbed problem, on its central path, and feasibility steps find strictly feasible iterates for the next perturbed problem. By using centering steps for the new perturbed problem, we obtain strictly feasible iterates close enough to the central path of the new perturbed problem. The procedure is repeated until an ?-approximate solution is found. We analyze the algorithm and obtain the complexity bound, which coincides with the best-known result for monotone linear complementarity problems.