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The ellipsoidal estimation of reachable sets is an efficient technique for the set-membership modelling of uncertain dynamical systems. In the paper, the optimal outer-ellipsoidal approximation of reachable sets is considered, and attention is paid to the criterion associated with the projection of the approximating ellipsoid onto a given direction. The nonlinear differential equations governing the evolution of ellipsoids are analyzed and simplified. The asymptotic behavior of the ellipsoids near the initial point and at infinity is studied. It is shown that the optimal ellipsoids under consideration touch the corresponding reachable sets at all time instants. A control problem for a system subjected to uncertain perturbations is investigated in the framework of the optimal ellipsoidal estimation of reachable sets. [ABSTRACT FROM AUTHOR]

This paper is concerned with a noncoercive unilateral problem in nonlinear elasticity with stored energy function having fast or slow growth. We prove an existence result for the associated minimization problem by using a recession approach in Orlicz-Sobolev space. Key Words. Nonlinear elasticity; Orlicz-Sobolev space; recession function, property (P). [ABSTRACT FROM AUTHOR]

Stochastic search methods for global optimization and multi-objective optimization are widely used in practice, especially on problems with black-box objective and constraint functions. Although there are many theoretical results on the convergence of stochastic search methods, relatively few deal with black-box constraints and multiple black-box objectives and previous convergence analyses require feasible iterates. Moreover, some of the convergence conditions are difficult to verify for practical stochastic algorithms, and some of the theoretical results only apply to specific algorithms. First, this article presents some technical conditions that guarantee the convergence of a general class of adaptive stochastic algorithms for constrained black-box global optimization that do not require iterates to be always feasible and applies them to practical algorithms, including an evolutionary algorithm. The conditions are only required for a subsequence of the iterations and provide a recipe for making any algorithm converge to the global minimum in a probabilistic sense. Second, it uses the results for constrained optimization to derive convergence results for stochastic search methods for constrained multi-objective optimization. [ABSTRACT FROM AUTHOR]

We study a class of convex multi-criteria optimization problems with convex objective functions under linear constraints. We use a non-scalarization method-namely, two implementable proximal point algorithms-to obtain the Pareto optimum under multi-criteria optimization. We show that the algorithms are globally convergent. We apply the algorithms to a supply chain risk management problem under multi-criteria considerations. [ABSTRACT FROM AUTHOR]

We give a necessary and sufficient condition for a difference of convex (DC, for short) functions, defined on a normed space, to be Lipschitz continuous. Our criterion relies on the intersection of the ε-subdifferentials of the involved functions. [ABSTRACT FROM AUTHOR]

We present the notion of weakly metrically regular functions on manifolds. Then, a sufficient condition for a real valued function defined on a complete Riemannian manifold to be weakly metrically regular is obtained, and two optimization problems on Riemannian manifolds are considered. Moreover, we present a generalization of the Palais-Smale condition for lower semicontinuous functions defined on manifolds. Then, we use this notion to obtain necessary conditions of optimality for a general minimization problem on complete Riemannian manifolds. [ABSTRACT FROM AUTHOR]

Dual characterizations of the containment of a convex set with quasiconvex inequality constraints are investigated. A new Lagrange-type duality and a new closed cone constraint qualification are described, and it is shown that this constraint qualification is the weakest constraint qualification for the duality. [ABSTRACT FROM AUTHOR]

We introduce the Hahn quantum variational calculus. Necessary and sufficient optimality conditions for the basic, isoperimetric, and Hahn quantum Lagrange problems, are studied. We also show the validity of Leitmann's direct method for the Hahn quantum variational calculus, and give explicit solutions to some concrete problems. To illustrate the results, we provide several examples and discuss a quantum version of the well known Ramsey model of economics. [ABSTRACT FROM AUTHOR]

A solution concept in optimization problems with interval-valued objective functions, which is essentially similar to the concept of nondominated solution in vector optimization problems, is introduced by imposing a partial ordering on the set of all closed intervals. The interval-valued Lagrangian function and interval-valued Lagrangian dual function are also proposed to formulate the dual problem of the interval-valued optimization problem. Under this setting, weak and strong duality theorems can be obtained. [ABSTRACT FROM AUTHOR]

A unified higher-order dual for a nondifferentiable minimax programming problem is formulated. Weak, strong and strict converse duality theorems are discussed involving generalized higher-order ( F, α, ρ, d)-Type I functions. [ABSTRACT FROM AUTHOR]

We establish the sufficient optimality conditions for a minimax programming problem involving p fractional n-set functions under generalized invexity. Using incomplete Lagrange duality, we formulate a mixed-type dual problem which unifies the Wolfe type dual and Mond-Weir type dual in fractional n-set functions under generalized invexity. Furthermore, we establish three duality theorems: weak, strong, and strict converse duality theorem, and prove that the optimal values of the primal problem and the mixed-type dual problem have no duality gap under extra assumptions in the framework. [ABSTRACT FROM AUTHOR]

We study first-order optimality conditions for the class of generalized semi-infinite programming problems (GSIPs). We extend various well-known constraint qualifications for finite programming problems to GSIPs and analyze the extent to which a corresponding Karush-Kuhn-Tucker (KKT) condition depends on these extensions. It is shown that in general the KKT condition for GSIPs takes a weaker form unless a certain constraint qualification is satisfied. In the completely convex case where the objective of the lower-level problem is concave and the constraint functions are quasiconvex, we show that the KKT condition takes a sharper form. [ABSTRACT FROM AUTHOR]

Given a set of points S = {x1, . . . , xm} ⊂ ℝn and ϵ > 0, we propose and analyze an algorithm for the problem of computing a (1 + ϵ )-approximation to the minimum-volume axis-aligned ellipsoid enclosing S.We establish that our algorithm is polynomial for fixed ϵ . In addition, the algorithm returns a small core set X ⊆ S, whose size is independent of the number of points m, with the property that the minimum-volume axis-aligned ellipsoid enclosing X is a good approximation of the minimum-volume axis-aligned ellipsoid enclosing S. Our computational results indicate that the algorithm exhibits significantly better performance than the theoretical worst-case complexity estimate. [ABSTRACT FROM AUTHOR]

We present a new constraint qualification which guarantees strong duality between a cone-constrained convex optimization problem and its Fenchel-Lagrange dual. This result is applied to a convex optimization problem having, for a given nonempty convex cone K, as objective function a K-convex function postcomposed with a K-increasing convex function. For this so-called composed convex optimization problem, we present a strong duality assertion, too, under weaker conditions than the ones considered so far. As an application, we rediscover the formula of the conjugate of a postcomposition with a K-increasing convex function as valid under weaker conditions than usually used in the literature. [ABSTRACT FROM AUTHOR]

We study multiobjective optimization problems with equilibrium constraints (MOPECs) described by parametric generalized equations in the form 0 ϵ G(x, y) + Q(x, y), where both mappings G and Q are set-valued. Such models arise particularly from certain optimization-related problems governed by variational inequalities and firstorder optimality conditions in nondifferentiable programming. We establish verifiable necessary conditions for the general problems under consideration and for their important specifications by using modern tools of variational analysis and generalized differentiation. The application of the obtained necessary optimality conditions is illustrated by a numerical example from bilevel programming with convex while nondifferentiable data. [ABSTRACT FROM AUTHOR]

This paper develops boundary integral representation formulas for the second variations of cost functionals for elliptic domain optimization problems. From the collection of all Lipschitz domains Ω which satisfy a constraint ∫Ω g(x) dx = 1, a domain is sought which maximizes either Fx0(Ω) = F(x0,u(x0)), fixed x0 ∈ Ω, or F(Ω) = ∫Ω F(x,u(x))dx, where u solves the Dirichlet problem Δu(x) = -f (x), x ∈ Ω, u(x) = 0, x ∈ ∂Ω. Necessary and sufficient conditions for local optimality are presented in terms of the first and second variations of the cost functionals Fx0 and F. The second variations are computed with respect to domain variations which preserve the constraint. After first summarizing known facts about the first variations of u and the cost functionals, a series of formulas relating various second variations of these quantities are derived. Calculating the second variations depends on finding first variations of solutions u when the data f are permitted to depend on the domain Ω. [ABSTRACT FROM AUTHOR]

In this paper, an algorithm for solving a mathematical programming problem with complementarity (or equilibrium) constraints (MPEC) is introduced, which uses the active-set methodology while maintaining the complementarity restrictions throughout the procedure. Finite convergence of the algorithm to a strongly stationary point of the MPEC is established under reasonable hypotheses. The algorithm can be easily implemented by adopting any active-set code for nonlinear programming. Computational experience is included to highlight the efficacy of the proposed method in practice. [ABSTRACT FROM AUTHOR]

We deal with the differential conditions for local optimality. The conditions that we derive for inequality constrained problems do not require constraint qualifications and are the broadest conditions based on only first-order and second-order derivatives. A similar result is proved for equality constrained problems, although the necessary conditions require the regularity of the equality constraints. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a method allowing us to choose the most suitable fuzzy implication in an inference system application. We introduce also a similarity measure, which we call degree of sameness of two fuzzy implications in an inference system application. [ABSTRACT FROM AUTHOR]

In this paper, we conduct three case studies to assess the effectiveness of a recently proposed first-order method for robust nonlinear programming [Zhang, Y.: J. Optim. Theory Appl. 132, 111-124 (2007)]. Three robust nonlinear programming problems were chosen from the literature using the criteria that results calculated using other methods must be available and the problems should be realistic, but fairly simple. Our studies show that the first-order method produced reasonable solutions when the level of uncertainty was small to moderate. In addition, we demonstrate a method for leveraging a theoretical result to eliminate constraint violations. Since the first-order method is relatively inexpensive in comparison to other robust optimization techniques, our studies indicate that, under moderate uncertainty, the first-order approach may be more suitable than other methods for large problems. [ABSTRACT FROM AUTHOR]

In the alternating directions method, the relaxation factor γ ϵ (0, √5+1―2 ) by Glowinski is useful in practical computations for structured variational inequalities. This paper points out that the same restriction region of the relaxation factor is also valid in the proximal alternating directions method. [ABSTRACT FROM AUTHOR]

Active constraint set invariancy sensitivity analysis is concerned with finding the range of parameter variation so that the perturbed problem has still an optimal solution with the same support set that the given optimal solution of the unperturbed problem has. However, in an optimization problem with inequality constraints, active constraint set invariancy sensitivity analysis aims to find the range of parameter variation, where the active constraints in a given optimal solution remains invariant. For the sake of simplicity, we consider the primal problem in standard form and consequently its dual may have an optimal solution with some active constraints. In this paper, the following question is answered: ‘what is the range of the parameter, where for each parameter value in this range, a dual optimal solution exists with exactly the same set of positive slack variables as for the current dual optimal solution?’. The differences of the results between the linear and convex quadratic optimization problems are highlighted too. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concepts of globally efficient solution and cone-Benson efficient solution for a vector equilibrium problem; we give some scalarization results for Henig efficient solution sets, globally efficient solution sets, weak efficient solution sets, and cone-Benson efficient solution sets in locally convex spaces. Using the scalarization results, we show the connectedness and path connectedness of weak efficient solution sets and various proper efficient solution sets of vector equilibrium problem. [ABSTRACT FROM AUTHOR]

In this note, we prove that the convergence results for vector optimization problems with equilibrium constraints presented in Wu and Cheng (J. Optim. Theory Appl. 125, 453–472, 2005) are not correct. Actually, we show that results of this type cannot be established at all. This is due to the possible lack, even under nice assumptions, of lower convergence of the solution map for equilibrium problems, already deeply investigated in Loridan and Morgan (Optimization 20, 819–836, 1989) and Lignola and Morgan (J. Optim. Theory Appl. 93, 575–596, 1997). [ABSTRACT FROM AUTHOR]

In this paper, we study the stabilization problem for a class of linear nonautonomous systems with norm-bounded controls. Using the Lyapunov function technique, we establish simple verifiable stabilizability conditions without solving any Riccati differential equation. Numerical examples are given to illustrate the results. [ABSTRACT FROM AUTHOR]

An approximation of a general V-ergodic semi-Markov game with Borel state space by discrete-state space strongly-ergodic games is studied. The standard expected ratio-average criterion as well as the expected time-average criterion are considered. New theorems on the existence of ϵ-equilibria are given. [ABSTRACT FROM AUTHOR]

The objective of this paper is to pose and solve a simple class of coordinated motion problems that capture the following question; When is formation travel optimal? This class of problems is complementary to two related classes of problems that have received significant attention: close formation travel and the generation of flocking behavior. We pose and solve this problem for the case of two agents traversing a finite, connected graph. Two examples are provided, illustrating the application of the results. [ABSTRACT FROM AUTHOR]

In this paper, we use existence theorems for the equilibria of generalized abstract economies proved recently to establish existence theorems for systems of generalized vector quasiequilibrium problems. Then, these existence theorems for equilibrium problems are used to derive existence theorems for systems of generalized vector quasivariational-like inequality problems, and vector quasioptimization problems. [ABSTRACT FROM AUTHOR]

Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering. [ABSTRACT FROM AUTHOR]

In this paper, we give counterexamples showing that the strong duality results obtained in Refs. 1-5 for several dual problems of multiobjective mathematical programs are false. We provide also the conditions under which correct results can be established. [ABSTRACT FROM AUTHOR]

We study the properties of the augmented Lagrangian function for nonlinear semidefinite programming. It is shown that, under a set of sufficient conditions, the augmented Lagrangian algorithm is locally convergent when the penalty parameter is larger than a certain threshold. An error estimate of the solution, depending on the penalty parameter, is also established. [ABSTRACT FROM AUTHOR]

In this paper, we reinvestigate the trust-region method by reformulating its subproblem: the trust-region radius is guided by gradient information at the current iteration and is self-adaptively adjusted. A trust-region algorithm based on the proposed subproblem is proved to be globally convergent. Moreover, the superlinear convergence of the new algorithm is shown without the condition that the Hessian of the objective function at the solution be positive definite. Preliminary numerical results indicate that the performance of the new method is notable. [ABSTRACT FROM AUTHOR]

We present a new duality theory to treat convex optimization problems and we prove that the geometric duality used by Scott and Jefferson in different papers during the last quarter of century is a special case of it. Moreover, weaker sufficient conditions to achieve strong duality are considered and optimality conditions are derived. Next, we apply our approach to some problems considered by Scott and Jefferson, determining their duals. We give weaker sufficient conditions to achieve strong duality and the corresponding optimality conditions. Finally, posynomial geometric programming is viewed also as a particular case of the duality approach that we present. [ABSTRACT FROM AUTHOR]

In this paper. the design problem of the dynamic output feedback controller for the asymptotic stabilization of a class of linear delay differential systems of the neutral type is considered. A criterion for the existence of such controller is derived based on the matrix inequality approach combined with the Lyapunov method. A parametrized characterization of the controller is given in terms of the feasible solutions to certain matrix inequalities, which can be solved by various convex optimization algorithms. A numerical example is given to illustrate the proposed design method. [ABSTRACT FROM AUTHOR]

One of the main ingredients of interior-point methods is the generation of iterates in a neighborhood of the central path. Measuring how close the iterates are to the central path is an important aspect of such methods and it is accomplished by using proximity measure functions. In this paper, we propose a unified presentation of the proximity measures and a study of their relationships and computational role when using a generic primal-dual interior-point method for computing the analytic center for a standard linear optimization problem. We demonstrate that the choice of the proximity measure can affect greatly the performance of the method. It is shown that we may be able to choose the algorithmic parameters and the central-path neighborhood radius (size) in such a way to obtain comparable results for several measures. We discuss briefly how to relate some of these results to nonlinear programming problems. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the determination of the entire set of Pareto solutions of location problems involving Q general criteria. These criteria include median, center, or centdian objective functions as particular instances. We characterize the set of Pareto solutions of all these multicriteria problems for any polyhedral gauge. An efficient algorithm is developed for the planar case and its complexity is established. Extensions to the nonconvex case are also considered. The proposed approach is more general than previously published approaches to multicriteria location problems. [ABSTRACT FROM AUTHOR]

The function fΔg: (x, y) ↦ g(y)f(x/g(y)), y ∊ dom g, is jointly convex provided f is convex and nonpositive at the origin and provided g is concave and nonnegative on its effective domain. Its convex conjugate combines the convex conjugates of f and -g by means of the same composition law. The effective domain of f Δg is then studied, which will prove to be useful in Part 2 of this paper (algebraic properties, Ref 1). [ABSTRACT FROM AUTHOR]

This paper deals with the class of uncertain continuous- time linear stochastic hybrid systems with Wiener process. The uncertainties that we are considering are of the norm-bounded type. The robust H∞ stochastic stabilization problem is treated. LMIs based sufficient conditions are developed to design the state feedback controller that robustly and stochastically stabilizes the studied class of systems and at the same time rejects a disturbance of desired level. The minimum disturbance rejection is also determined. A numerical example is provided to show the validity of the proposed results. [ABSTRACT FROM AUTHOR]

We study the problem of computing a (1 + ϵ)-approximation to the minimum-volume enclosing ellipsoid of a given point set S = {p¹, p², ¡, pn} ⊆ Rd Based on a simple, initial volume approximation method, we propose a modification of the Khachiyan first-order algorithm. Our analysis leads to a slightly improved complexity bound of O(nd³/ϵ) operations for ϵ ∊ (0, 1). As a byproduct, our algorithm returns a core set X ⊆ S with the property that the minimum-volume enclosing ellipsoid of X provides a good approximation to that of S. Furthermore, the size of X depends on only the dimension d and ϵ, but not on the number of points n. In particular, our results imply that ∣X∣ = O(d²/ϵ) for ϵ ∊(0, 1). [ABSTRACT FROM AUTHOR]

In this paper, we study the dynamics of a differential inclusion built upon a nonsmooth, not necessarily convex, constrained minimization problem in finite-dimensional spaces. In particular, we are interested in the investigation of the asymptotic behavior of the trajectories of the dynamical system represented by the differential inclusion. Under suitable assumptions on the constraint set and the two involved functions (one defining the constraint set, the other representing the functional to be minimized), it is proved that all the trajectories converge to the set of the constrained critical points. We present also a large class of constraint sets satisfying our assumptions. As a simple consequence. in the case of a smooth convex minimization problem, we have that any trajectory converges to the set of minimizers. [ABSTRACT FROM AUTHOR]

Mathematical programs with equilibrium constraints (MPEC) are nonlinear programs which do not satisfy any of the common constraint qualifications (CQ). In order to obtain first-order optimality conditions, constraint qualifications tailored to the MPECs have been developed and researched in the past. In this paper. we introduce a new Abadie-type constraint qualification for MPECs. We investigate sufficient conditions for this new CQ. discuss its relationship to several existing MPEC constraint qualifications, and introduce a new Slater-type constraint qualifications. Finally, we prove a new stationarity concept to be a necessary optimality condition under our new Abadie-type CQ. [ABSTRACT FROM AUTHOR]

In this paper, a model of a linear multilevel programming problem with dominated objective functions (LMPPD(l)) is proposed, where multiple reactions of the lower levels do not lead to any uncertainty in the upper-level decision making. Under the assumption that the constrained set is nonempty and bounded, a necessary optimality condition is obtained. Two types of geometric properties of the solution sets are studied. It is demonstrated that the feasible set of LMPPD(l) is neither necessarily composed of faces of the constrained set nor necessarily connected. These properties are different from the existing theoretical results for linear multilevel programming problems. [ABSTRACT FROM AUTHOR]

An analytic center cutting-plane method with deep cuts for semidefinite feasibility problems is presented. Our objective in these problems is to find a point in a nonempty bounded convex set r in the cone of symmetric positive-semidefinite matrices. The cutting plane method achieves this by the following iterative scheme. At each iteration, a query point Ŷ that is an approximate analytic center of the current working set is chosen. We assume that there exists an oracle which either confirms that Ŷ ∈ Γ or returns a cut A ∈ Sm such that {Y ∈ Sm:A · Y ≤ A · Ŷ - ξ} ⊃ Γ, where ξ ≥ 0. If Ŷ Γ, an approximate analytic center of the new working set, defined by adding the new cut to the preceding working set, is then computed via a primal Newton procedure. Assuming that Γ contains a ball with radius ∈ > 0, the algorithm obtains eventually a point in Γ, with a worst-case complexity of O&zast;(m3/∈2) on the total number of cuts generated. [ABSTRACT FROM AUTHOR]

Defining speed of diffusion as the amount of time it takes to get from one penetration level to a higher one, we introduce a dynamic model in which we study the link between pricing policy, speed of diffusion, and number of competitors in the market. Our analysis shows that, in the case of strategic (oligopolistic) competition, the speed of diffusion has an important influence on the optimal pricing policy. In particular, we find that higher speeds of diffusion create an incentive to strategically interacting firms to lower their prices. [ABSTRACT FROM AUTHOR]

In this paper, the property of a necessary second-order optimality condition to hold with the same Lagrange multiplier for all critical vectors is investigated. It is limited to nonconvex optimization problems in &IRlig;n with equality and inequality constraints; the Mangasarian-Fromovitz constraint qualification is assumed to hold. A counterexample was given recently by Anitescu. We give some sufficient conditions and we prove that this property holds if n ≤ 2 or if the number of active inequality constraints is at most two. For three active inequality constraints and n = 3, a counterexample is given. [ABSTRACT FROM AUTHOR]

A portfolio optimization problem on an infinite-time horizon is considered. Risky asset prices obey a logarithmic Brownian motion and interest rates vary according to an ergodic Markov diffusion process. The goal is to choose optimal investment and consumption policies to maximize the infinite-horizon expected discounted hyperbolic absolute risk aversion (HARA) utility of consumption. The problem is then reduced to a one-dimensional stochastic control problem by virtue of the Girsanov transformation. A dynamic programming principle is used to derive the dynamic programming equation (DPE). The subsolution/supersolution method is used to obtain existence of solutions of the DPE. The solutions are then used to derive the optimal investment and consumption policies. In addition, for a special case, we obtain the results using the viscosity solution method. [ABSTRACT FROM AUTHOR]

This work is concerned with the maximum principle for optimal control problem governed by magnetohydrodynamic equations, which describe the motion of a viscous incompressible conducting fluid in a magnetic field and consist of a subtle coupling of the Navier-Stokes equation of a viscous incompressible fluid flow and the Maxwell equation of an electromagnetic field. An integral type state constraint is considered. [ABSTRACT FROM AUTHOR]

We prove a sufficient global optimality condition for the problem of minimizing a quadratic function subject to quadratic equality constraints where the variables are allowed to take values -1 and 1. We extend the condition to quadratic problems with matrix variables and orthonormality constraints, and in particular to the quadratic assignment problem. [ABSTRACT FROM AUTHOR]

In this paper, we propose a nonmonotone trust-region algorithm for the solution of optimization problems with general nonlinear equality constraints and simple bounds. Under a constant rank assumption on the gradients of the active constraints, we analyze the global convergence of the proposed algorithm. [ABSTRACT FROM AUTHOR]

This paper develops a theory for the global solution of nonconvex optimization problems with parameter-embedded linear dynamic systems. A quite general problem formulation is introduced and a solution is shown to exists. A convexity theory for integrals is then developed to construct convex relaxations for utilization in a branch-and-bound framework to calculate a global minimum. Interval analysis is employed to generate bounds on the state variables implied by the bounds on the embedded parameters. These bounds, along with basic integration theory, are used to prove convergence of the branch-and-bound algorithm to the global minimum of the optimization problem. The implementation of the algorithm is then considered and several numerical case studies are examined thoroughly. [ABSTRACT FROM AUTHOR]