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.     

Second order necessary conditions in set constrained differentiable vector optimization

We state second order necessary optimality conditions for a vector optimization problem with an arbitrary feasible set and an order in the final space given by a pointed convex cone with nonempty interior. We establish, in finite-dimensional spaces, second order optimality conditions in dual form by means of Lagrange multipliers rules when the feasible set is defined by a function constrained to a set with convex tangent cone. To pass from general conditions to Lagrange multipliers rules, a generalized Motzkin alternative theorem is provided. All the involved functions are assumed to be twice Fréchet differentiable. Mathematics subject classification 2000:90C29, 90C46This research was partially supported by Ministerio de Ciencia y Tecnología (Spain), project BMF2003-02194.     

First and second order sufficient conditions for strict minimality in nonsmooth vector optimization

In this paper we present first and second order sufficient conditions for strict local minima of orders 1 and 2 to vector optimization problems with an arbitrary feasible set and a twice directionally differentiable objective function. With this aim, the notion of support function to a vector problem is introduced, in such a way that the scalar case and the multiobjective case, in particular, are contained. The obtained results extend the multiobjective ones to this case. Moreover, specializing to a feasible set defined by equality, inequality, and set constraints, first and second order sufficient conditions by means of Lagrange multiplier rules are established.     

Optimality conditions in mathematical programming and composite optimization

New second order optimality conditions for mathematical programming problems and for the minimization of composite functions are presented. They are derived from a general second order Fermat's rule for the minimization of a function over an arbitrary subset of a Banach space. The necessary conditions are more accurate than the recent results of Kawasaki (1988) and Cominetti (1989); but, more importantly, in the finite dimensional case they are twinned with sufficient conditions which differ by the replacement of an inequality by a strict inequality. We point out the equivalence of the mathematical programming problem with the problem of minimizing a composite function. Our conditions are especially important when one deals with functional constraints. When the cone defining the constraints is polyhedral we recover the classical conditions of Ben-Tal—Zowe (1982) and Cominetti (1990).     

Necessary and sufficient conditions for quadratic minimality

We provide necessary and sufficient conditions for a (non-convex) quadratic function to take a local minimum over a convex set. Various limiting examples are given.     

Second order parabolic equations in Banach spaces with dynamic boundary conditions

In this paper, we exhibit a unified treatment of the mixed initial boundary value problem for second order (in time) parabolic linear differential equations in Banach spaces, whose boundary conditions are of a dynamical nature. Results regarding existence, uniqueness, continuous dependence (on initial data) and regularity of classical and strict solutions are established. Moreover, several examples are given as samples for possible applications.

     

Impulsive BVPs with nonlinear boundary conditions for the second order differential equations without growth restrictions

The paper deals with the impulsive nonlinear boundary value problem

     

Second order sufficient conditions for optimizing with equality constraints

The paper establishes the equivalence to second order of the Jacobean (constrained gradient) and the Lagrangean approaches to optimizing a function subject to equality constraints. The second order equivalence does not seem to have been considered previously. It is established using a new second order sufficiency condition which has computational advantages over the other two.     

零对称BZ-代数的充要条件

重新证明文[10]中几个重要结论并修正文[10]中的定理1(11)和定理2．在此基础上，利用这些重新证明过的结论及修正过的定理可以按照文[10]中引理3，定理4，定理6，定理7，定理10的证明过程原样证明文[10]中的相应结果．因而在文[10]中，除性质11是结合BZ一代数的等价性质(见文[15])，定理1(11)及定理2需要进行修正外，其余结论及证明过程均成立．     

Second order differential operators with Feller-Wentzell type boundary conditions

In this paper, we study a basic generation problem concerning the second order differential operator in the space C[0,1] of complex continuous functions equipped with Feller-Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468-519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C[0,1] (or on a subspace of C[0,1]), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.     

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

In this paper, we give second-order necessary and sufficient optimality conditions for a minimization problem of a sup-type functionS(x)=sup{f(x,t);t T}, whereT is a compact set in a metric space and f is a function defined on ⁿ ×T. Our conditions are stated in terms of the first and second derivatives of f(x, t) with respect tox, and involve an extra term besides the second derivative of the ordinary Lagrange function. The extra term is essential when {f(x,t)} _t forms an envelope. We study the relationship between our results, Wetterling [14], and Hettich and Jongen [6].     

On necessary and sufficient conditions in vector optimization

It is shown that some general multiplier rules are necessary conditions for vector optimization in infinite-dimensional spaces. Under additional convexity assumptions, these conditions are sufficient. As an application, the Pontryagin maximum principle for cooperative differential games is examined.The authors are grateful to Professor W. Stadler and the referees of the previous edition of this paper for their valuable remarks and suggestions, which have been very helpful in the preparation of this paper.     

Coefficient bounds for some families of starlike and convex functions of complex order

In the present work, the authors determine coefficient bounds for functions in certain subclasses of starlike and convex functions of complex order, which are introduced here by means of a family of nonhomogeneous Cauchy–Euler differential equations. Several corollaries and consequences of the main results are also considered.     

Some necessary and sufficient conditions for the convergence of sequences of unilateral convex sets

We establish some necessary and sufficient conditions for the convergence in $\text{W}{}_0{}^{\mathrm{1,p}}\text{(Ω)}$ of sequences of convex sets determined by obstacles. These conditions are expressed in terms of properties of the p-capacities of the level sets of the obstacles.     

Structural interpretation and derivation of necessary and sufficient conditions for delay moments in FIFO multiserver queues

Scheller-Wolf [12] established necessary and sufficient conditions for finite stationary delay moments in stable FIFO GI/GI/s queues that incorporate the interaction between service time distribution, traffic intensity (ρ) and the number of servers in the queue. These conditions can be used to show that when the service time has finite first but infinite αth moment, s slow servers can give lower delays than one fast server. In this paper, we derive an alternative derivation of these moment results: Both upper bounds, that serve as sufficient conditions, and lower bounds, that serve as necessary conditions are presented. In addition, we extend the class of service time distributions for which the necessary conditions are valid. Our new derivations provide a structural interpretation of the moment bounds, giving intuition into their origin: We show that FIFO GI/GI/s delay can be represented as the minimum of (s − k) i.i.d. GI/GI/1 delays, when ρ satisfies k < ρ < k+1. AMS Subject Classification 60K25     

The second order optimality conditions for nonlinear mathematical programming with C 1,1 data

To find nonlinear minimization problems are considered and standard C ²-regularity assumptions on the criterion function and constrained functions are reduced to C ^1,1-regularity. With the aid of the generalized second order directional derivative for C ^1,1 real-valued functions, a new second order necessary optimality condition and a new second order sufficient optimality condition for these problems are derived.     

KKT conditions for weak compact convex sets,theorems of the alternative,and optimality conditions

We present several equivalent conditions for the Karush–Kuhn–Tucker conditions for weak^? compact convex sets. Using them, we extend several existing theorems of the alternative in terms of weak^? compact convex sets. Such extensions allow us to express the KKT conditions and hence necessary optimality conditions for more general nonsmooth optimization problems with inequality and equality constraints. Furthermore, several new equivalent optimality conditions for optimization problems with inequality constraints are obtained.     

Second order optimality conditions for generalized semi-infinite programming problems

We consider generalized semi-infinite programming problems. Second order necessary and sufficient conditionsfor local optimality are given. The conditions are derived under assumptions such that the feasible set can be described by means of a finite number of optimal value functions. Since we do not require a strict complementary condition for the local reduction these functions are only of class C¹ A sufficient condition for optimality is proven under much weaker assumptions.     

On sufficient second order optimality conditions in multiobjective optimization

A second order sufficient optimality criterion is presented for a multiobjective problem subject to a constraint given just as a set. To this aim, we first refine known necessary conditions in such a way that the sufficient ones differ by the replacement of inequalities by strict inequalities. Furthermore, we show that no relationship holds between this criterion and a sufficient multipliers rule, when the constraint is described by inequalities and equalities. Finally, improvements of this criterion for the unconstrained case are presented, stressing the differences with single-objective optimization