Adjoint Coexhausters in Nonsmooth Analysis and Extremality Conditions

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.

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.

Some second-order necessary conditions in optimal control

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.

This is a survey of high‐order necessary optimality conditions for control problems with inequality and equality terminal constraints. Both classical and Pontryagin extremals are considered. For classical extremals, the characteristics of necessary optimality conditions in terms of variations are given, and the investigations devoted to Kelley's conditions are discussed in detail. For Pontryagin extremals, two types of results are given: generalizations and extensions of matrix impulse optimality condition and high‐order maximum principle.

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.

A class of nonsmooth vector optimization problems are considered, where the feasible set defined by cone constraint and the objective and constraint functions are locally Lipschitz. The concept of the Clarke's generalized directional derivative for a locally Lipschitz function is introduced. By using this concept, necessary optimality condition for the unconstrained optimization problem is established. Furthermore, a Slater-type constraint qualification is given in such a way that they generalize the classical one, when the constraint functions are differentiable. Then, Kuhn–Tucker necessary optimality condition in terms of the Clarke subdifferentials is obtained.

The field-dependent relaxation dynamics of suspended magnetic nanoparticles continues to present a fascinating topic of basic science that at the same time is highly relevant for several technological and biomedical applications. Renewed interest in the intriguing behavior of magnetic nanoparticles in response to external fields has at least in parts be driven by rapid advances in magnetic fluid hyperthermia research. Although a wealth of experimental, theoretical, and simulation studies have been performed in this field in recent years, several contradictory findings have so far prevented the emergence of a consistent picture. Here, we present a dynamic mean-field theory together with comprehensive computer simulations of a microscopic model system to systematically discuss the influence of several key parameters on the relaxation dynamics, such as steric and dipolar interactions, the external magnetic field strength and frequency, as well as the ratio of Brownian and Néel relaxation time. We also discuss the specific and intrinsic loss power as measures of the efficiency of magnetic fluid heating and discuss optimality conditions in terms of fluid and field parameters. Our results are helpful to reconcile contradictory findings in the literature and provide an important step towards a more consistent understanding. In addition, our findings also help to select experimental conditions that optimize magnetic fluid heating applications.

In this paper, wavelet techniques are employed for the fast numerical solution of a control problem governed by an elliptic boundary value problem with boundary control. A quadratic cost functional involving natural norms of the state and the control is to be minimized. Firstly the constraint, the elliptic boundary value problem, is formulated in an appropriate weak form that allows to handle varying boundary conditions explicitly: the boundary conditions are treated by Lagrange multipliers, leading to a saddle point problem. This is combined with a fictitious domain approach in order to cover also more complicated boundaries. Deviating from standard approaches, we then use (biorthogonal) wavelets to derive an equivalent infinite discretized control problem which involves only e2-norms and -operators. Classical methods from optimization yield the corresponding optimality conditions in terms of two weakly coupled (still infinite) saddle point problems for which a unique solution exists. For deriving finite-dimensional systems which are uniformly invertible, stability of the discretizations has to be ensured. This together with the e2-setting circumvents the problem of preconditioning: all operators have uniformly bounded condition numbers independent of the discretization. In order to numerically solve the resulting (finite-dimensional) linear system of the weakly coupled saddle point problems, a fully iterative method is proposed which can be viewed as an inexact gradient scheme. It consists of a gradient algorithm as an outer iteration which alternatingly picks the two saddle point problems, and an inner iteration to solve each of the saddle point problems, exemplified in terms of the Uzawa algorithm. It is proved here that this strategy converges, provided that the inner systems are solved sufficiently well. Moreover, since the system matrix is well-conditioned, it is shown that in combination with a nested iteration strategy this iteration is asymptotically optimal in the sense that it provides the solution on discretization level J with an overall amount of arithmetic operations that is proportional to the number of unknows NJ on that level. Finally, numerical results are provided.

The subgradient, under strict efficiency, of a set-valued mapping is developed, and the existence of the subgradient is proved. Optimality conditions in terms of Lagrange multipliers for a strictly efficient point are established in the general case and in the case with ic-cone-convexlike data.

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.

In this chapter, the authors develop an optimal control model of knowledge dissemination among people in the society. The knowledge transfer system is formulated in term of compartmental model, where the society members are categorized into four classes based on knowledge acquisition and their willingness to disseminate. The model is equipped with a set of control variables for process intervening, namely technical training for ignorant-immigrants, information dissemination through social media for solitariants and enthusiants, and technical training for solitariants. Optimality conditions in terms of differential equations system was derived by using Pontryagin minimum principle leading to the characterization of optimal control strategies that minimizing the number of solitariants, enthusiants, and ignorants simultaneously with the control efforts. The sweep method and the fourth order Runge-Kutta algorithm was implemented to numerically solve the equation systems. The effectiveness of the control strategies toward a set of control scenarios was evaluated through examples.

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.

The aim of this paper is to present optimality conditions for Pareto efficiency of some set-valued optimization problems by means of Bouligand derivatives. The framework is that of general Banach spaces and the non-emptiness of the interior of the ordering cone in the output space is not assumed.

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings $$ (VP)\left\{\matrix{{\mathbb R}_{+}^{p} &-\hbox{Minimize} f(x)-g(x),\cr &\hbox{subject to the constraints} \cr & x\in C,l(x)\in -Q,Ax=b\cr &\hbox{and } h(x)-k(x)\in -{\mathbb R}_{+}^{m},}\right. $$ where \(f:=(f_{1},\ldots,f_{p}),g:=(g_{1},\ldots,g_{p}) h:=(h_{1},\ldots,h_{m}), k:=(k_{1},\ldots,k_{m}), Q\) is a closed convex cone in a Banach space Z, l is a mapping Q-convex from a Banach space X into Z, A is a continuous linear operator from X into a Banach space \(W, {\mathbb R}_{+}^{p}\) and \({\mathbb R}_{+}^{m}\) are respectively the nonnegative orthants of \({\mathbb R}^{p}\) and \({\mathbb R}^{m}\), C is a nonempty closed convex subset of X, b∈W, and the functions f i ,g i ,h j and k j are convex for i=1,...,p and j=1,ldots,m. Necessary optimality conditions for (VP) are established in terms of Lagrange-Fritz-John multipliers. When the set of constraints for (VP) is convex and under the generalized Slater constraint qualification introduced in Jeyakumar and Wolkowicz [11] , we derive necessary optimality conditions in terms of Lagrange-Karush-Kuhn-Tucker multipliers which are also sufficient whenever the functions g i ,i=1,...,p are polyhedrals. Our approach consists in using a special scalarization function. A necessary optimality condition for convex vector maximization problem is derived. Also an application to vector fractional mathematical programming is given. Our contribution extends the results obtained in scalar optimization by Hiriart-Urruty [9] and improve substantially the few results known in vector case (see for instance: [11], [12] and [14]).