On necessary optimality conditions in vector optimization problems

Necessary conditions of the multiplier rule type for vector optimization problems in Banach spaces are proved by using separation theorems and Ljusternik's theorem. The Pontryagin maximum principle for multiobjective control problems with state constraints is derived from these general conditions. The paper extends to vector optimization results established in the scalar case by Ioffe and Tihomirov.     

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.     

On strong and weak second-order necessary optimality conditions for nonlinear programming

Second-order necessary optimality conditions play an important role in optimization theory. This is explained by the fact that most numerical optimization algorithms reduce to finding stationary points satisfying first-order necessary optimality conditions. As a rule, optimization problems, especially the high dimensional ones, have a lot of stationary points so one has to use second-order necessary optimality conditions to exclude nonoptimal points. These conditions are closely related to second-order constraint qualifications, which guarantee the validity of second-order necessary optimality conditions. In this paper, strong and weak second-order necessary optimality conditions are considered and their validity proved under so-called critical regularity condition at local minimizers.     

Approximate necessary conditions for locally weak Pareto optimality

Necessary conditions for a given pointx ⁰ to be a locally weak solution to the Pareto minimization problem of a vector-valued functionF=(f ₁,...,f _m ),F:XR ^m,XR ^m, are presented. As noted in Ref. 1, the classical necessary condition-conv {Df ₁(x ⁰)|i=1,...,m}T ^*(X, x ⁰) need not hold when the contingent coneT is used. We have proven, however, that a properly adjusted approximate version of this classical condition always holds. Strangely enough, the approximation form>2 must be weaker than form=2.The authors would like to thank the anonymous referee for the suggestions which led to an improved presentation of the paper.     

Refinements of necessary conditions for optimality in nonlinear programming

In this paper, a new set of necessary conditions for optimality is introduced with reference to the differentiable nonlinear programming problem. It is shown that these necessary conditions are sharper than the usual Fritz John ones. A constraint qualification relevant to the new necessary conditions is defined and extensions to the locally Lipschitz case are presented.     

Second order necessary optimality conditions for minimizing a sup-type function

The second derivative of an envelope cannot be expressed only by second derivatives of the constituent functions. By taking account of this fact, we derive new second order necessary optimality conditions for minimization of a sup-type function. The conditions involve an extra term besides the second derivative of the Lagrange function. Furthermore, we will comment on the relationship between the extra term and a kind of second order directional derivative of the sup-type function.     

No-gap optimality conditions for an optimal control problem with pointwise control-state constraints

An optimal control problem with pointwise mixed constraints of the instationary three-dimensional Navier–Stokes–Voigt equations is considered. We derive second-order optimality conditions and show that there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, the second-order sufficient optimality conditions for the problem where the objective functional does not contain a Tikhonov regularization term are also discussed.     

Necessary optimality conditions for singular controls

A definition of singular controls with respect to components is given that includes, in particular, the conventional definition. On the basis of this definition, new necessary optimiality conditions for singular controls with respect to components are derived for the processes governed by systems of ordinary differential equations.     

On sufficient optimality conditions for a quasiconvex programming problem

Some remarks are made on a paper by Bector, Chandra, and Bector (see Ref. 1) concerning the Fritz John and Kuhn-Tucker sufficient optmality conditions as well as duality theorems for a nonlinear programming problem with a quasiconvex objective function.This research was supported by the Italian Ministry of University Scientific and Technological Research.     

Generalized Bellman-Hamilton-Jacobi optimality conditions for a control problem with a boundary condition

In this paper we study conditions for optimality of a deterministic control problem where the state of the system is required to stop at the boundary. Using the Clarke generalized gradient, we refine the classical verification theorem and show that it is not only sufficient but also necessary for optimality. It is also shown that the solution to the generalized Bellman-Jacobi-Hamilton equation involving the Clarke generalized gradient is unique among the class of regular functions.     

Additional necessary conditions for optimal control with state-variable inequality constraints

It is shown that, when the set of necessary conditions for an optimal control problem with state-variable inequality constraints given by Bryson, Denham, and Dreyfus is appropriately augmented, it is equivalent to the (different) set of conditions given by Jacobson, Lele, and Speyer. Relationships among the various multipliers are given.This work was done at NASA Ames Research Center, Moffett Field, California, under a National Research Council Associateship.     

Coderivative necessary optimality conditions for sharp and robust efficiencies in vector optimization with variable ordering structure

In this paper, we define two new concepts of efficiency for vector optimization with variable ordering structure, namely the sharp and robust efficiencies, and we study their connections with classical concepts of efficiency in vector optimization. Then, we get necessary optimality conditions for them using Fréchet and Mordukhovich calculus coupled with the Gerstewitz’s (Tammer’s) scalarizing functional and openness results for set-valued maps.     

一类变结构动态系统的非光滑最优性条件

研究了一类事件驱动的变结构动态系统的非光滑最优性条件. 通过引入一个新的时间变量, 将变结构动态系统的最优性问题转化为古典动态系统的最优性问题. 基于广义微分和古典动态系统的最优性理论, 得到了该系统的Frechet上微分形式的必要性条件, 推广了已有文献的相关结论. 结果表明, 在系统的连续运行过程中, 控制变量、协态变量和状态变量满足最小值原理和协态方程. 在系统的运行模型发生改变时, 协态变量产生一定的跳跃, 哈密尔顿函数连续. 最后通过一个算例说明了该结论的有效性.     

Realistic modelled optimal control problems in aerospace engineering - a challenge to the necessary optimality

A control problem for a hypersonic space vehicle is used to illustrate the need for a generalization of the necessary optimality conditions in the accurate numerical solution of more realistic models for optimal control problems in aerospace engineering.     

Necessary conditions of Pareto optimality for multiobjective optimal control problems under constraints

This paper studies multiobjective optimal control problems in presence of constraints in the discrete time framework. Both the finite- and infinite-horizon settings are considered. The paper provides necessary conditions of Pareto optimality under lighter smoothness assumptions compared to the previously obtained results. These conditions are given in the form of weak and strong Pontryagin principles which generalize the existing ones. To obtain some of these results, we provide new multiplier rules for multiobjective static optimization problems and new Pontryagin principles for the finite horizon multiobjective optimal control problems.     

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.     

Necessary optimality conditions of a D.C. set-valued bilevel optimization problem

In this article, we consider a bilevel vector optimization problem where objective and constraints are set valued maps. Our approach consists of using a support function [1–3,14,15,32] together with the convex separation principle for the study of necessary optimality conditions for D.C. bilevel set-valued optimization problems. We give optimality conditions in terms of the strong subdifferential of a cone-convex set-valued mapping introduced by Baier and Jahn 6 Baier, J and Jahn, J. 1999. On subdifferentials of set-valued maps. J. Optim. Theory Appl., 100: 233–240. [Crossref], [Web of Science ®] , [Google Scholar] and the weak subdifferential of a cone-convex set-valued mapping of Sawaragi and Tanino 28 Sawaragi, Y and Tanino, T. 1980. Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl., 31: 473–499.  [Google Scholar]. The bilevel set-valued problem is transformed into a one level set-valued optimization problem using a transformation originated by Ye and Zhu 34 Ye, JJ and Zhu, DL. 1995. Optimality conditions for bilevel programming problems. Optimization, 33: 9–27. [Taylor & Francis Online] , [Google Scholar]. An example illustrating the usefulness of our result is also given.     

Sufficient optimality conditions and duality for a quasiconvex programming problem

Under differentiability assumptions, Fritz John Sufficient optimality conditions are proved for a nonlinear programming problem in which the objective function is assumed to be quasiconvex and the constraint functions are assumed to quasiconcave/strictly pseudoconcave. Duality theorems are proved for Mond-Weir type duality under the above generalized convexity assumptions.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are thankful to Professor B. Mond for suggestions that improved the original draft of the paper.