On optimality conditions in nonsmooth inequality constrained minimization

First order necessary optimality conditions for a minimum of an inequality constrained minimization problem are given in terms of approximate quasidifferentials, without the usual differentiability, convexity or locally Lipschitz assumptions. The main result is obtained with the help of a semi-infinite Gordan type alternative theorem. Sufficient conditions for a minimum are also given with the usual convexity assumption replaced by an invex condition.     

Second-order optimality conditions for cone-subarcwise connected set-valued optimization problems

The concept of a cone subarcwise connected set-valued map is introduced. Several examples are given to illustrate that the cone subarcwise connected set-valued map is a proper generalization of the cone arcwise connected set-valued map, as well as the arcwise connected set is a proper generalization of the convex set, respectively. Then, by virtue of the generalized second-order contingent epiderivative, second-order necessary optimality conditions are established for a point pair to be a local global proper efficient element of set-valued optimization problems. When objective function is cone subarcwise connected, a second-order sufficient optimality condition is also obtained for a point pair to be a global proper efficient element of set-valued optimization problems.     

Second-order necessary conditions for general problems with state inequality constraints

Journal of Optimization Theory and Applications - This paper is a sequel to a previous article by the author, concerned with a certain canonical problem in optimal control involving constraints of...     

Second-order necessary optimality conditions for optimization problems involving set-valued maps

Second-order necessary optimality conditions are established under a regularity assumption for a problem of minimizing a functiong over the solution set of an inclusion system 0 F(x), x M, whereF is a set-valued map between finite-dimensional spaces andM is a given subset. The proof of the main result of the paper is based on the theory of infinite systems of linear inequalities.     

Second-order necessary optimality conditions for semilinear elliptic optimal control problems

This paper deals with the necessary optimality conditions for semilinear elliptic optimal control problems with a pure pointwise state constraint and mixed pointwise constraints. By computing the so-called ‘sigma-term’, we obtain the second-order necessary optimality conditions for the problems, which is sharper than some previously established results in the literature. Besides, we give a condition which relaxes the Slater condition and guarantees that the Lagrangian is normalized.     

New necessary conditions of optimality for control problems with state-variable inequality constraints

Necessary conditions of optimality for state-variable inequality constrained problems are derived which differ from those of Bryson, Denham, and Speyer with regard to the behavior of the adjoint variables at junctions of interior and boundary arcs. In particular, it is shown that the earlier conditions under-specify the behavior of the adjoint variables at the junctions. An example is used to demonstrate that the earlier conditions may yield non-stationary trajectories.     

Necessary optimality conditions for infinite dimensional state constrained control problems

This paper is concerned with first order necessary optimality conditions for state constrained control problems in separable Banach spaces. Assuming inward pointing conditions on the constraint, we give a simple proof of Pontryagin maximum principle, relying on infinite dimensional neighboring feasible trajectories theorems proved in [20]. Further, we provide sufficient conditions guaranteeing normality of the maximum principle. We work in the abstract semigroup setting, but nevertheless we apply our results to several concrete models involving controlled PDEs. Pointwise state constraints (as positivity of the solutions) are allowed.     

A trust region method for solving linearly constrained locally Lipschitz optimization problems

In this paper, we present a nonsmooth trust region method for solving linearly constrained optimization problems with a locally Lipschitz objective function. Using the approximation of the steepest descent direction, a quadratic approximation of the objective function is constructed. The null space technique is applied to handle the constraints of the quadratic subproblem. Next, the CG-Steihaug method is applied to solve the new approximation quadratic model with only the trust region constraint. Finally, the convergence of presented algorithm is proved. This algorithm is implemented in the MATLAB environment and the numerical results are reported.     

Second-order optimality conditions for nondominated solutions of multiobjective programming with C 1,1 data

We examine new second-order necessary conditions and sufficient conditions which characterize nondominated solutions of a generalized constrained multiobjective programming problem. The vector-valued criterion function as well as constraint functions are supposed to be from the class C ^1,1. Second-order optimality conditions for local Pareto solutions are derived as a special case.     

Second-order sequence-based necessary optimality conditions in constrained nonsmooth vector optimization and applications

Several notions of sequential directional derivatives and sequential local approximations are introduced. Under (first-order) Hadamard differentiability assumptions of the data at the point of study, these concepts are utilized to analyze second-order necessary optimality conditions, which rely on given sequences, for local weak solutions in nonsmooth vector optimization problems with constraints. Some applications to minimax programming problems are also derived.     

Second-order necessary optimality conditions for domain optimization problems of the Neumann type

In this paper, we treat a domain optimization problem in which the boundary-value problem is a Neumann problem. In the case where the domain is in a three-dimensional Euclidean space, the first-order and the second-order necessary conditions which the optimal domain must satisfy are derived under a constraint which is the generalization of the requisite of constant volume.Portions of this paper were presented at the 13th IFIP Conference on System Modelling and Optimization, Tokyo, Japan, 1987.     

Necessary optimality conditions for constrained optimization problems under relaxed constraint qualifications

We derive first- and second-order necessary optimality conditions for set-constrained optimization problems under the constraint qualification-type conditions significantly weaker than Robinson’s constraint qualification. Our development relies on the so-called 2-regularity concept, and unifies and extends the previous studies based on this concept. Specifically, in our setting constraints are given by an inclusion, with an arbitrary closed convex set on the right-hand side. Thus, for the second-order analysis, some curvature characterizations of this set near the reference point must be taken into account.      

Second-order optimality conditions for minimizing a max-function

We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.     

Second-order optimality conditions for minimizing a maxfunction

We study Chaney's and Ben-Tal-Zowe's second-order directional derivatives with applications in minimization problem for max-functions of the formh(x): = max {f(x, τ); τ ∈T},where T is a compact metricspace. We improve Kawasaki's result on necessary condition for such functions in the minimization problem.     

Second-order optimality conditions using approximations for nonsmooth vector optimization problems under inclusion constraints

Second-order necessary conditions and sufficient conditions for optimality in nonsmooth vector optimization problems with inclusion constraints are established. We use approximations as generalized derivatives and avoid even continuity assumptions. Convexity conditions are not imposed explicitly. Not all approximations in use are required to be bounded. The results improve or include several recent existing ones. Examples are provided to show that our theorems are easily applied in situations where several known results do not work.     

A nondegenerate fuzzy optimality condition for constrained optimization problems without qualification conditions

Fuzzy optimization conditions in terms of the Fréchet subdifferential for reflexive spaces were investigated by Borwein, Treiman and Zhu (1998) in [1]. To achieve the nondegenerate form, it is well known that some qualification conditions should be assumed. In this paper, we are going to prove that the nondegenerate fuzzy optimality condition even holds with no qualification conditions in Asplund spaces (in particular, reflexive spaces) for optimization problems with semi-continuous and continuous data. The results are even new in finite-dimensional frameworks.     

Second-order optimality principle for singular optimal control problems

A new necessary condition for singular optimal control problems is presented in this paper. The condition is simpler to apply than existing conditions and is easily derived from a Taylor series expansion of the performance index.