Optimality Conditions In Terms Research Articles

Dimension reduction for thin films prestrained by shallow curvature

We are concerned with the dimension reduction analysis for thin three-dimensional elastic films, prestrained via Riemannian metrics with weak curvatures. For the prestrain inducing the incompatible version of the Föppl–von Kármán equations, we find the Γ -limits of the rescaled energies, identify the optimal energy scaling laws, and display the equivalent conditions for optimality in terms of both the prestrain components and the curvatures of the related Riemannian metrics. When the stretching-inducing prestrain carries no in-plane modes, we discover similarities with the previously described shallow shell models. In higher prestrain regimes, we prove new energy upper bounds by constructing deformations as the Kirchhoff–Love extensions of the highly perturbative, Hölder-regular solutions to the Monge–Ampere equation obtained by means of convex integration.

Optimality Conditions for a Nonsmooth Semivectorial Bilevel Optimization Problem

In this paper, we are concerned with a nonsmooth semivectorial bilevel optimization problem (P). Using a scalarization technique, we transform the lower-level problem into several scalar optimization problems We give necessary optimality condition in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first-order necessary optimality conditions are then derived in the special case where all functions are strictly differentiable.

Higher-order efficiency conditions for constrained vector equilibrium problems

ABSTRACT In this paper, we investigate some optimality conditions of higher-order for nonsmooth nonconvex vector equilibrium problems with constraints (or without constraints) in terms of the higher-order upper and lower Studniarski derivatives in Banach spaces. The calculus rule of all the data involved in the problem is taken into account. Using the notion of higher-order upper Studniarski derivative with the class of m-stable/m-steady functions, we first provide the higher-order necessary optimality conditions for the local weak efficient solution of vector equilibrium problem without constraints and then we present the higher-order necessary and sufficient optimality conditions for the local strict minimum of order m to such problem. We second obtain the necessary and sufficient optimality conditions for those efficient solutions of vector equilibrium problem with cone and equality constraints through Lagrange multiplier rules in finite-dimensional spaces. Final, the higher-order necessary and sufficient optimality conditions in terms of the higher-order upper and lower Studniarski derivatives for the local weak efficient solution of vector equilibrium problem with set, cone and equality constraints in Banach spaces are also established. Some examples are proposed to demonstrate our findings.

Necessary optimality conditions for a fractional multiobjective optimization problem

In this paper, we are concerned with a fractional multiobjective optimization problem (P). Using support functions together with a generalized Guignard constraint qualification, we give necessary optimality conditions in terms of convexificators and the Karush–Kuhn–Tucker multipliers. Several intermediate optimization problems have been introduced to help us in our investigation.

No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation

This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-to-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish “no-gap” second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided.

A Second-Order Sufficient Optimality Condition for Risk-Neutral Bi-level Stochastic Linear Programs

The expectation functionals, which arise in risk-neutral bi-level stochastic linear models with random lower-level right-hand side, are known to be continuously differentiable, if the underlying probability measure has a Lebesgue density. We show that the gradient may fail to be local Lipschitz continuous under this assumption. Our main result provides sufficient conditions for Lipschitz continuity of the gradient of the expectation functional and paves the way for a second-order optimality condition in terms of generalized Hessians. Moreover, we study geometric properties of regions of strong stability and derive representation results, which may facilitate the computation of gradients.

Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential

In the research of optimization problems, optimality conditions play an important role. By using some derivatives, various types of necessary and/or sufficient optimality conditions have been introduced by many researchers. Especially, in convex programming, necessary and sufficient optimality conditions in terms of the subdifferential have been studied extensively. Recently, necessary and sufficient optimality conditions for quasiconvex programming have been investigated by the authors. However, there are not so many results concerned with Karush–Kuhn–Tucker type optimality conditions for non-differentiable quasiconvex programming. In this paper, we study a Karush–Kuhn–Tucker type optimality condition for quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. We show some closedness properties for Greenberg–Pierskalla subdifferential. Under the Slater constraint qualification, we show a necessary and sufficient optimality condition for essentially quasiconvex programming in terms of Greenberg–Pierskalla subdifferential. Additionally, we introduce a necessary and sufficient constraint qualification of the optimality condition. As a corollary, we show a necessary and sufficient optimality condition for convex programming in terms of the subdifferential.

Shape turnpike for linear parabolic PDE models

We introduce and study the turnpike property for time-varying shapes, within the viewpoint of optimal control. We focus here on second-order linear parabolic equations where the shape acts as a source term and we seek the optimal time-varying shape that minimizes a quadratic criterion. We first establish existence of optimal solutions under some appropriate sufficient conditions. We then provide necessary conditions for optimality in terms of adjoint equations and, using the concept of strict dissipativity, we prove that state and adjoint satisfy the measure-turnpike property, meaning that the extremal time-varying solution remains essentially close to the optimal solution of an associated static problem. We show that the optimal shape enjoys the exponential turnpike property in terms of Hausdorff distance for a Mayer quadratic cost. We illustrate the turnpike phenomenon in optimal shape design with several numerical simulations.

Optimality conditions for a multiobjective bilevel optimization problem involving set valued constraints

ABSTRACT In this paper, we investigate a multiobjectif bilevel programming problem with set valued constraints in both the upper problem and the lower problem. Using the notion of the support function to the feasible set mapping, together with an appropriate constraint qualification, we establish necessary optimality conditions in terms of asymptotical pointwise compact approximations; continuity of data functions and compactness of approximations are not necessarily required. In order to get sufficient optimality conditions, we introduce generalized pseudo-invexity and quasi-invexity notions in terms of asymptotical pointwise compact approximations. Examples that illustrate our results are given.

Relaxed constant positive linear dependence constraint qualification and its application to bilevel programs

Relaxed constant positive linear dependence constraint qualification (RCPLD) for a system of smooth equalities and inequalities is a constraint qualification that is weaker than the usual constraint qualifications such as Mangasarian Fromovitz constraint qualification and the linear constraint qualification. Moreover RCPLD is known to induce an error bound property. In this paper we extend RCPLD to a very general feasibility system which may include Lipschitz continuous inequality constraints, complementarity constraints and abstract constraints. We show that this RCPLD for the general system is a constraint qualification for the optimality condition in terms of limiting subdifferential and limiting normal cone and it is a sufficient condition for the error bound property under the strict complementarity condition for the complementarity system and Clarke regularity conditions for the inequality constraints and the abstract constraint set. Moreover we introduce and study some sufficient conditions for RCPLD including the relaxed constant rank constraint qualification. Finally we apply our results to the bilevel program.

Optimality conditions in terms of convexificators for a bilevel multiobjective optimization problem

ABSTRACT In this paper, we are concerned with a multiobjective bilevel optimization problem , where the lower-level problem is also a multiobjective optimization problem. Using the optimal-value reformulation, together with a scalarization technique, under a nonsmooth generalized Guignard constraint qualification, one gives necessary optimality conditions for the initial problem. As an application, one deduces optimality conditions for a scalar bilevel programming problem in terms of convexificators. An example is given to illustrate our findings.

Infinite-dimensional vector optimization and a separation theorem

ABSTRACT Using relative minimality solutions for vector optimization problems, we establish optimality conditions in terms of Karush–Kuhn–Tucker multipliers as well as Fritz-John multipliers. Moreover, we give a general separation theorem including both finite and infinite classical separation theorem.

A New Constraint Qualification and Sharp Optimality Conditions for Nonsmooth Mathematical Programming Problems in Terms of Quasidifferentials

The paper is devoted to an analysis of a new constraint qualification and a derivation of the strongest existing optimality conditions for nonsmooth mathematical programming problems with equality and inequality constraints in terms of Demyanov-Rubinov-Polyakova quasidifferentials under the minimal possible assumptions. To this end, we obtain a novel description of convex subcones of the contingent cone to a set defined by quasidifferentiable equality and inequality constraints with the use of a new constraint qualification. We utilize these description and constraint qualification to derive the strongest existing optimality conditions for nonsmooth mathematical programming problems in terms of quasidifferentials under less restrictive assumptions than in previous studies. The main feature of the new constraint qualification and related optimality conditions is the fact that they depend on individual elements of quasidifferentials of the objective function and constraints and are not invariant with respect to the choise of quasidifferentials. To illustrate the theoretical results, we present two simple examples in which optimality conditions in terms of various subdifferentials (in fact, any outer semicontinuous/limiting subdifferential) are satisfied at a nonoptimal point, while the optimality conditions obtained in this paper do not hold true at this point, that is, optimality conditions in terms of quasidifferentials, unlike the ones in terms of subdifferentials, detect the nonoptimality of this point.

Optimality Conditions for Approximate Pareto Solutions of a Nonsmooth Vector Optimization Problem with an Infinite Number of Constraints

In this paper, we present some new necessary and sufficient optimality conditions in terms of Clarke subdifferentials for approximate Pareto solutions of a nonsmooth vector optimization problem which has an infinite number of constraints. As a consequence, we obtain optimality conditions for the particular cases of cone-constrained convex vector optimization problems and semidefinite vector optimization problems. Examples are given to illustrate the obtained results.

Optimality conditions for a bilevel optimization problem in terms of KKT multipliers and convexificators

In this paper we investigate a bilevel optimization problem by using the optimistic approach. Under a non smooth generalized Guignard constraint qualification, due the optimal value reformulation, the necessary optimality conditions in terms of convexificators and Karush-Kuhn-Tucker (KKT) multipliers are given.

Necessary optimality conditions for a semivectorial bilevel optimization problem using the kth-objective weighted-constraint approach

In this paper, we have pointed out that the proof of Theorem 11 in the recent paper (Lafhim in Positivity, 2019. https://doi.org/10.1007/s11117-019-00685-1 ) is erroneous. Using techniques from variational analysis, we propose other proofs to detect necessary optimality conditions in terms of Karush–Kuhn–Tucker multipliers. Our main results are given in terms of the limiting subdifferentials and the limiting normal cones. Completely detailed first order necessary optimality conditions are then given in the smooth setting while using the generalized differentiation calculus of Mordukhovich.

Optimal Control of a Phase Field System Modelling Tumor Growth with Chemotaxis and Singular Potentials

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fr\'echet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

On Lagrange Multiplier Rules for Set-Valued Optimization Problems in the Sense of Set Criterion

In this paper, we obtain a new scalar representation of set-valued optimization problems by using a criterion of solution introduced by Kuroiwa, this criterion is a weaker version of Pareto efficient solution. Following this scalarization, we establish necessary optimality conditions in terms of Fritz John multipliers as well as Karush-Kuhn-Tucker multipliers. Moreover, we give sufficient optimality conditions.

Metric Regularity of Quasidifferentiable Mappings and Optimality Conditions for Nonsmooth Mathematical Programming Problems

This article is devoted to the analysis of necessary and/or sufficient conditions for metric regularity in terms of Demyanov-Rubinov-Polyakova quasidifferentials. We obtain new necessary and sufficient conditions for the local metric regularity of a multifunction in terms of quasidifferentials of the distance function to this multifunction. We also propose a new MFCQ-type constraint qualification for a parametric system of quasidifferentiable equality and inequality constraints and prove that it ensures the metric regularity of a multifunction associated with this system. As an application, we utilize our constraint qualification to strengthen existing optimality conditions for quasidifferentiable programming problems with equality and inequality constraints. We also prove the independence of the optimality conditions of the choice of quasidifferentials and present a simple example in which the optimality conditions in terms of quasidifferentials detect the non-optimality of a given point, while optimality conditions in terms of various subdifferentials fail to disqualify this point as non-optimal.

Subdifferentials of Value Functions in Nonconvex Dynamic Programming for Nonstationary Stochastic Processes

The main goal of this paper is to apply the machinery of variational analysis and generalized differentiation to study infinite horizon stochastic dynamic programming (DP) with discrete time in the Banach space setting without convexity assumptions. Unlike to standard stochastic DP with stationary Markov processes, we investigate here stochastic DP in $L^p$ spaces to deal with nonstationary stochastic processes, which describe a more flexible learning procedure for the decision-maker. Our main concern is to calculate generalized subgradients of the corresponding value function and to derive necessary conditions for optimality in terms of the stochastic Euler inclusion under appropriate Lipschitzian assumptions. The usage of the subdifferential formula for integral functionals on $L^p$ spaces allows us, in particular, to find verifiable conditions to ensure smoothness of the value function without any convexity and/or interiority assumptions.