一类不可微广义分式规划的K-T型必要条件

本文对一类在Rn的开子集X上的非线性不等式约束的广义分式规划问题: 目标函数中的分子是可微函数与凸函数之和而分母是可微函数与凸函数之差,且约束函数是可微的,在Abadie约束品性或Calmness约束品性下,给出了最优解的Kuhn-Tucker 型必要条件,所得结果改进和推广了已有文献中的相应结果．     

On the optimal control problems with Gâteaux differentiable operator constraints

In this paper the problem of optimal control with mixed equality and inequality operator constraints is considered under the assumption of Gâteaux differentiability. The extremum principle for this kind of problem is obtained by using some specification of the Dubovitskii-Milyutin method for the case of n equality constraints given by Gâteaux differentiable operators.     

Semismooth Newton method for the lifted reformulation of mathematical programs with?complementarity constraints

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 BD-regular 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. Preliminary numerical results are also reported.     

Calmness of partially perturbed linear systems with an application to the central path

ABSTRACT

In this paper we develop point-based formulas for the calmness modulus of the feasible set mapping in the context of linear inequality systems with a fixed abstract constraint and (partially) perturbed linear constraints. The case of totally perturbed linear systems was previously analyzed in [Cánovas MJ, López MA, Parra J, et al. Calmness of the feasible set mapping for linear inequality systems. Set-Valued Var Anal. 2014;22:375–389, Section 5]. We point out that the presence of such an abstract constraint yields the current paper to appeal to a notable different methodology with respect to previous works on the calmness modulus in linear programming. The interest of this model comes from the fact that partially perturbed systems naturally appear in many applications. As an illustration, the paper includes an example related to the classical central path construction. In this example we consider a certain feasible set mapping whose calmness modulus provides a measure of the convergence of the central path. Finally, we underline the fact that the expression for the calmness modulus obtained in this paper is (conceptually) implementable as far as it only involves the nominal data.     

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.     

The effect of calmness on the solution set of systems of nonlinear equations

We address the problem of solving a continuously differentiable nonlinear system of equations under the condition of calmness. This property, also called upper Lipschitz-continuity in the literature, can be described by a local error bound and is being widely used as a regularity condition in optimization. Indeed, it is known to be significantly weaker than classic regularity assumptions that imply that solutions are isolated. We prove that under this condition, the rank of the Jacobian of the function that defines the system of equations must be locally constant on the solution set. In addition, we prove that locally, the solution set must be a differentiable manifold. Our results are illustrated by examples and discussed in terms of their theoretical relevance and algorithmic implications.     

Constrained Consumptions, Lipschitzian Demands, and Regular Economies

We consider an exchange economy where the consumers face linear inequality constraints on consumption. We parametrize the economy with the initial endowments and constraints. We exhibit sufficient conditions on the constraints implying that the demand is locally Lipschitzian and continuously differentiable on an open dense subset of full Lebesgue measure. Using this property, we show that the equilibrium manifold is lipeomorphic to an open, connected subset of an Euclidean space and that the lipeomorphism is almost everywhere continuously differentiable. We prove that regular economies are generic and that they have a finite odd number of equilibrium prices and local differentiable selections of the equilibrium prices.Communicated by J. P. CrouzeixThis work was partially supported by CCE, ECOS, and ICM Sistemas Complejos de Ingeniería.     

On second-order Fritz John type optimality conditions in nonsmooth multiobjective programming

We study a multiobjective optimization program with a feasible set defined by equality constraints and a generalized inequality constraint. We suppose that the functions involved are Fréchet differentiable and their Fréchet derivatives are continuous or stable at the point considered. We provide necessary second order optimality conditions and also sufficient conditions via a Fritz John type Lagrange multiplier rule and a set-valued second order directional derivative, in such a way that our sufficient conditions are close to the necessary conditions. Some consequences are obtained for parabolic directionally differentiable functions and C ^1,1 functions, in this last case, expressed by means of the second order Clarke subdifferential. Some illustrative examples are also given.     

Constraint qualifications for constrained Lipschitz optimization problems and applications to a MPCC

Three constraint qualifications (the weak generalized Robinson constraint qualification, the bounded constraint qualification, and the generalized Abadie constraint qualification), which are weaker than the generalized Robinson constraint qualification (GRCQ) given by Yen (1997) [1], are introduced for constrained Lipschitz optimization problems. Relationships between those constraint qualifications and the calmness of the solution mapping are investigated. It is demonstrated that the weak generalized Robinson constraint qualification and the bounded constraint qualification are easily verifiable sufficient conditions for the calmness of the solution mapping, whereas the proposed generalized Abadie constraint qualification, described in terms of graphical derivatives in variational analysis, is weaker than the calmness of the solution mapping. Finally, those constraint qualifications are written for a mathematical program with complementarity constraints (MPCC), and new constraint qualifications ensuring the C-stationary point condition of a MPCC are obtained.     

A viability approach to the inverse set-valued map theorem

The purpose of this paper is to characterize by means of viability tools the pseudo-lipschitzianity property of a set-valued map F in a neighborhood of a point of its graph in terms of derivatives of this set-valued map F in a neighborhood of a point of its graph, instead of using the transposes of the derivatives. On the way, we relate these properties to the calmness index of a set-valued map, an extensions of Clarke’s calmness of a function, as well as Doyen’s Lipschitz kernel of a set-valued map, which is the largest Lipschitz submap.     

Exact penalty functions and calmness for mathematical programming under nonlinear perturbations

In the present paper, the effects of nonlinear perturbations of constraint systems are considered over the relationship between calmness and exact penalization, within the context of mathematical programming with equilibrium constraints. Two counterexamples are provided showing that the crucial link between the existence of penalty functions and the property of calmness for perturbed problems is broken in the presence of general perturbations. Then, some properties from variational analysis are singled out, which are able to restore to a certain extent the broken link. Consequently, conditions on the value function associated to perturbed optimization problems are investigated in order to guarantee the occurrence of the above properties.     

Primal and dual optimality criteria in convex programming

Summary This paper considers the problem of minimizing a convex differentiable function subject to convex differentiable constraints. Necessary and sufficient conditions (not requiring any constraints qualification) for a point to be an optimal solution are given in terms of a parametric linear program. Dual characterization theorems are then derived, which generalizes the classical results ofKuhn-Tucker andJohn.

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Zusammenfassung Es wird das Problem betrachtet, eine konvexe differenzierbare Funktion unter konvexen differenzierbaren Nebenbedingungen zu minimieren. Unter Verwendung eines parametrischen linearen Optimierungsproblems werden notwendige und hinreichende Bedingungen für Optimalität eines Punktes angegeben, die keine constraints qualifications benötigen. Sodann werden duale Charakterisierungstheoreme hergeleitet, welche die klassischen Resultate vonKuhn-Tucker undJohn verallgemeinern.

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This research was partly supported by Project No. NR 047-021, ONR Contract N00014-75-C-0569 with the Center for Cybernetic Studies, The University of Texas. Reproduction in whole or in part is permitted for any purpose of the United States Government.     

Injectivity of differentiable maps ℝ2 → ℝ2 at infinity

The main result given in Theorem 1.1 is a condition for a map X, defined on the complement of a disk D in ℝ² with values in ℝ², to be extended to a topological embedding of ℝ², not necessarily surjective. The map X is supposed to be just differentiable with the condition that, for some ε > 0, at each point the eigenvalues of the differential do not belong to the real interval (-ε,∞). The extension is obtained by restricting X to the complement of some larger disc. The result has important connections with the property of asymptotic stability at infinity for differentiable vector fields.     

Calmness and Exact Penalization in Vector Optimization with Cone Constraints

In this paper, a (local) calmness condition of order α is introduced for a general vector optimization problem with cone constraints in infinite dimensional spaces. It is shown that the (local) calmness is equivalent to the (local) exact penalization of a vector-valued penalty function for the constrained vector optimization problem. Several necessary and sufficient conditions for the local calmness of order α are established. Finally, it is shown that the local calmness of order 1 implies the existence of normal Lagrange multipliers. Presented at the 6th International Conference on Optimization: Techniques and Applications, Ballarat, Australia, December 9–11, 2004 This work is supported by the Postdoctoral Fellowship of Hong Kong Polytechnic University.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

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.     

A note on upper Lipschitz stability, error bounds, and critical multipliers for Lipschitz-continuous KKT systems

We prove a new local upper Lipschitz stability result and the associated local error bound for solutions of parametric Karush–Kuhn–Tucker systems corresponding to variational problems with Lipschitzian base mappings and constraints possessing Lipschitzian derivatives, and without any constraint qualifications. This property is equivalent to the appropriately extended to this nonsmooth setting notion of noncriticality of the Lagrange multiplier associated to the primal solution, which is weaker than second-order sufficiency. All this extends several results previously known only for optimization problems with twice differentiable data, or assuming some constraint qualifications. In addition, our results are obtained in the more general variational setting.     

Stability Analysis for Parameterized Variational Systems with Implicit Constraints

In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples.

     

Contingent cone to a set defined by equality and inequality constraints at a Fréchet differentiable point

In this paper, we prove equality expression for the contingent cone and the strict normal cone to a set determined by equality and/or inequality constraints at a Fréchet differentiable point. A similar result has appeared before in the literature under the assumption that all the constraint functions are of classC or under the assumption that the functions are strictly differentiable at the point in question. Our result has applications to the calculation of various kinds of tangent cones and normal cones.This research was supported, in part by the National Science and Engineering Research Council of Canada under Grant No. OGP-41983.The authors would like to thank D. E. Ward for his many helpful comments.     

On necessary conditions for a class of nondifferentiable minimax fractional programming

The Kuhn–Tucker-type necessary optimality conditions are given for the problem of minimizing a max fractional function, where the numerator of the function involved is the sum of a differentiable function and a convex function while the denominator is the difference of a differentiable function and a convex function, subject to a set of differentiable nonlinear inequalities on a convex subset C$C$ of Rⁿ${\mathbb{R}}^n$, under the conditions similar to the Kuhn–Tucker constraint qualification or the Arrow–Hurwicz–Uzawa constraint qualification or the Abadie constraint qualification. Relations with the calmness constraint qualification are given.