Stability of semi-infinite vector optimization problems without compact constraints

This paper is devoted to the continuity of solution maps for perturbation semi-infinite vector optimization problems without compact constraint sets. The sufficient conditions for lower semicontinuity and upper semicontinuity of solution maps under functional perturbations of both objective functions and constraint sets are established. Some examples are given to analyze the assumptions in the main result.     

On regularity and stability in semi-infinite optimization

For stationary solutions and Langrange multipliers of a semi-infinite program withC ¹ data, we study some stability behaviour which is closely related to (metric) regularity of the constraint system. The multiplier set mapping considered here has its images in a finite-dimensional space. In this framework, regularity is a necessary and sufficient condition to have bounded sets of multipliers.     

Clarke coderivatives of efficient point multifunctions in parametric vector optimization

The paper is devoted to the study of the Clarke/circatangent coderivatives of the efficient point multifunction of parametric vector optimization problems in Banach spaces. We provide inner/outer estimates for evaluating the Clarke/circatangent coderivative of this multifunction in a broad class of conventional vector optimization problems in the presence of geometrical, operator and (finite and infinite) functional constraints. Examples are given for analyzing and illustrating the obtained results.     

Variational analysis of functions of the roots of polynomials

The Gauss-Lucas Theorem on the roots of polynomials nicely simplifies the computation of the subderivative and regular subdifferential of the abscissa mapping on polynomials (the maximum of the real parts of the roots). This paper extends this approach to more general functions of the roots. By combining the Gauss-Lucas methodology with an analysis of the splitting behavior of the roots, we obtain characterizations of the subderivative and regular subdifferential for these functions as well. In particular, we completely characterize the subderivative and regular subdifferential of the radius mapping (the maximum of the moduli of the roots). The abscissa and radius mappings are important for the study of continuous and discrete time linear dynamical systems. Dedicated to R. Tyrrell Rockafellar on the occasion of his 70th birthday. Terry is one of those rare individuals who combine a broad vision, deep insight, and the outstanding writing and lecturing skills crucial for engaging others in his subject. With these qualities he has won universal respect as a founding father of our discipline. We, and the broader mathematical community, owe Terry a great deal. But most of all we are personally thankful to Terry for his friendship and guidance. Research supported in part by the National Science Foundation Grant DMS-0203175. Research supported in part by the Natural Sciences and Engineering Research Council of Canada. Research supported in part by the National Science Foundation Grant DMS-0412049.     

Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces

In this paper, we first introduce the concept of Levitin-Polyak well-posedness of a generalized mixed variational inequality in Banach spaces and establish some characterizations of its Levitin-Polyak well-posedness. Under suitable conditions, we prove that the Levitin-Polyak well-posedness of a generalized mixed variational inequality is equivalent to the Levitin-Polyak well-posedness of a corresponding inclusion problem and a corresponding fixed point problem. We also derive some conditions under which a generalized mixed variational inequality in Banach spaces is Levitin-Polyak well-posed.     

A semi-infinite programming algorithm for solving optimal power flow with transient stability constraints

This paper proposes a new algorithm for solving a type of complicated optimal power flow (OPF) problems in power systems, i.e., OPF problems with transient stability constraints (OTS). The OTS is converted into a semi-infinite programming (SIP) via some suitable function analysis. Then based on the KKT system of the reformulated SIP, a smoothing quasi-Newton algorithm is presented in which the numerical integration is used. The convergence of the algorithm is established. An OTS problem in power system is tested, which shows that the proposed algorithm is promising.     

Sufficient conditions for pseudo-Lipschitz property in convex semi-infinite vector optimization problems

This paper is devoted to the study of the pseudo-Lipschitz property of the efficient (Pareto) solution map for the perturbed convex semi-infinite vector optimization problem (CSVO). We establish sufficient conditions for the pseudo-Lipschitz property of the efficient solution map of (CSVO) under continuous perturbations of the right-hand side of the constraints and functional perturbations of the objective function. Examples are given to illustrate the obtained results.     

On Hölder continuity of approximate solutions to parametric equilibrium problems

We establish verifiable sufficient conditions for Hölder continuity of approximate solutions to parametric equilibrium problems, when solutions may be not unique. Many examples are provided to illustrate the need of considering approximate solutions instead of exact solutions and the essentialness of the imposed assumptions. As applications, we derive this Hölder continuity for constrained minimization, variational inequalities and fixed point problems.     

Well-posed equilibrium problems

In this paper we introduce some notions of well-posedness for scalar equilibrium problems in complete metric spaces or in Banach spaces. As equilibrium problem is a common extension of optimization, saddle point and variational inequality problems, our definitions originates from the well-posedness concepts already introduced for these problems.We give sufficient conditions for two different kinds of well-posedness and show by means of counterexamples that these have no relationship in the general case. However, together with some additional assumptions, we show via Ekeland’s principle for bifunctions a link between them.Finally we discuss a parametric form of the equilibrium problem and introduce a well-posedness concept for it, which unifies the two different notions of well-posedness introduced in the first part.     

Tikhonov-type regularization method for efficient solutions in vector optimization

The paper is devoted to developing the Tikhonov-type regularization algorithm of finding efficient solutions to the vector optimization problem for a mapping between finite dimensional Hilbert spaces with respect to the partial order induced by a pointed closed convex cone. We prove that under some suitable conditions either the sequence generated by our method converges to an efficient solution or all of its cluster points belong to the set of all efficient solutions of this problem.     

Levitin-Polyak well-posedness of variational inequalities

In this paper we consider the Levitin-Polyak well-posedness of variational inequalities. We derive a characterization of the Levitin-Polyak well-posedness by considering the size of Levitin-Polyak approximating solution sets of variational inequalities. We also show that the Levitin-Polyak well-posedness of variational inequalities is closely related to the Levitin-Polyak well-posedness of minimization problems and fixed point problems. Finally, we prove that under suitable conditions, the Levitin-Polyak well-posedness of a variational inequality is equivalent to the uniqueness and existence of its solution.     

信赖域策略的内点投影既约Hessian方法解非线性约束优化

A interior point scaling projected reduced Hessian method with combination of nonmonotonic backtracking technique and trust region strategy for nonlinear equality constrained optimization with nonegative constraint on variables is proposed. In order to deal with large problems,a pair of trust region subproblems in horizontal and vertical subspaces is used to replace the general full trust region subproblem. The horizontal trust region subproblem in the algorithm is only a general trust region subproblem while the vertical trust region subproblem is defined by a parameter size of the vertical direction subject only to an ellipsoidal constraint. Both trust region strategy and line search technique at each iteration switch to obtaining a backtracking step generated by the two trust region subproblems. By adopting the l1 penalty function as the merit function, the global convergence and fast local convergence rate of the proposed algorithm are established under some reasonable conditions. A nonmonotonic criterion and the second order correction step are used to overcome Maratos effect and speed up the convergence progress in some ill-conditioned cases.     

A spatial domain decomposition method for parabolic optimal control problems

We present a non-overlapping spatial domain decomposition method for the solution of linear–quadratic parabolic optimal control problems. The spatial domain is decomposed into non-overlapping subdomains. The original parabolic optimal control problem is decomposed into smaller problems posed on space–time cylinder subdomains with auxiliary state and adjoint variables imposed as Dirichlet boundary conditions on the space–time interface boundary. The subdomain problems are coupled through Robin transmission conditions. This leads to a Schur complement equation in which the unknowns are the auxiliary state adjoint variables on the space-time interface boundary. The Schur complement operator is the sum of space–time subdomain Schur complement operators. The application of these subdomain Schur complement operators is equivalent to the solution of an subdomain parabolic optimal control problem. The subdomain Schur complement operators are shown to be invertible and the application of their inverses is equivalent to the solution of a related subdomain parabolic optimal control problem. We introduce a new family of Neumann–Neumann type preconditioners for the Schur complement system including several different coarse grid corrections. We compare the numerical performance of our preconditioners with an alternative approach recently introduced by Benamou.     

Uniform subsmoothness and linear regularity for a collection of infinitely many closed sets

Motivated by the subsmoothness of a closed set introduced by Aussel et al. (2005) [8], we introduce and study the uniform subsmoothness of a collection of infinitely many closed subsets in a Banach space. Under the uniform subsmoothness assumption, we provide an interesting subdifferential formula on distance functions and consider uniform metric regularity for a kind of multifunctions frequently appearing in optimization and variational analysis. Different from the existing works, without the restriction of convexity, we consider several fundamental notions in optimization such as the linear regularity, CHIP, strong CHIP and property (G) for a collection of infinitely many closed sets. We establish relationships among these fundamental notions for an arbitrary collection of uniformly subsmooth closed sets. In particular, we extend duality characterizations of the linear regularity for a collection of closed convex sets to the nonconvex setting.     

Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints

This paper deals with the optimal control problem of an ordinary differential equation with several pure state constraints, of arbitrary orders, as well as mixed control-state constraints. We assume (i) the control to be continuous and the strengthened Legendre–Clebsch condition to hold, and (ii) a linear independence condition of the active constraints at their respective order to hold. We give a complete analysis of the smoothness and junction conditions of the control and of the constraints multipliers. This allows us to obtain, when there are finitely many nontangential junction points, a theory of no-gap second-order optimality conditions and a characterization of the well-posedness of the shooting algorithm. These results generalize those obtained in the case of a scalar-valued state constraint and a scalar-valued control.     

Higher-order Mond–Weir duality for set-valued optimization

In this paper, we introduce a higher-order Mond–Weir dual for a set-valued optimization problem by virtue of higher-order contingent derivatives and discuss their weak duality, strong duality and converse duality properties.     

Characterization of total ill-posedness in linear semi-infinite optimization

This paper deals with the stability of linear semi-infinite programming (LSIP, for short) problems. We characterize those LSIP problems from which we can obtain, under small perturbations in the data, different types of problems, namely, inconsistent, consistent unsolvable, and solvable problems. The problems of this class are highly unstable and, for this reason, we say that they are totally ill-posed. The characterization that we provide here is of geometrical nature, and it depends exclusively on the original data (i.e., on the coefficients of the nominal LSIP problem). Our results cover the case of linear programming problems, and they are mainly obtained via a new formula for the subdifferential mapping of the support function.     

A superlinearly convergent strongly sub-feasible SSLE-type algorithm with working set for nonlinearly constrained optimization

In this paper, by means of a new efficient identification technique of active constraints and the method of strongly sub-feasible direction, we propose a new sequential system of linear equations (SSLE) algorithm for solving inequality constrained optimization problems, in which the initial point is arbitrary. At each iteration, we first yield the working set by a pivoting operation and a generalized projection; then, three or four reduced linear equations with a same coefficient are solved to obtain the search direction. After a finite number of iterations, the algorithm can produced a feasible iteration point, and it becomes the method of feasible directions. Moreover, after finitely many iterations, the working set becomes independent of the iterates and is essentially the same as the active set of the KKT point. Under some mild conditions, the proposed algorithm is proved to be globally, strongly and superlinearly convergent. Finally, some preliminary numerical experiments are reported to show that the algorithm is practicable and effective.     

Fixed point optimization algorithm and its application to network bandwidth allocation

A convex optimization problem for a strictly convex objective function over the fixed point set of a nonexpansive mapping includes a network bandwidth allocation problem, which is one of the central issues in modern communication networks. We devised an iterative algorithm, called a fixed point optimization algorithm, for solving the convex optimization problem and conducted a convergence analysis on the algorithm. The analysis guarantees that the algorithm, with slowly diminishing step-size sequences, weakly converges to a unique solution to the problem. Moreover, we apply the proposed algorithm to a network bandwidth allocation problem and show its effectiveness.