Journal of Optimization Theory and Applications最新文献

A Fan-Theobald-von Neumann system [7] is a triple ((mathcal {V},mathcal {W},lambda )), where (mathcal {V}) and (mathcal {W}) are real inner product spaces and (lambda :mathcal {V}rightarrow mathcal {W}) is a norm-preserving map satisfying a Fan-Theobald-von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decomposition systems (Eaton triples). The present article is a continuation of [9] where the concepts of commutativity, automorphisms, majorization, and reduction were introduced and elaborated. Here, we describe some transfer principles and present Fenchel conjugate and subdifferential formulas.

The present paper deals with a maximal monotone operator A of type (D) in a Banach space whose dual space is strictly convex. We establish some representations for the value Ax at a given point x via its values at nearby points of x. We show that the faces of Ax are contained in the set of all weak(^*) convergent limits of bounded nets of the operator at nearby points of x, then we obtain a representation for Ax by use of this set. In addition, representations for the support function of Ax based on the minimal-norm selection of the operator in certain Banach spaces are given.

We study the generalized sequential normal compactness in variational analysis and establish characterizations of the property of graphs of weakly differentiable mappings between Banach spaces, as well as calculus rules involving such functions.

In this paper, we propose and analyze a third-order dynamical system for finding zeros of the sum of two generalized operators in a Hilbert space (mathcal {H}). We establish the existence and uniqueness of the trajectories generated by the system under appropriate continuity conditions, and prove exponential convergence to the unique zero when the sum of the operators is strongly monotone. Additionally, we derive an explicit discretization of the dynamical system, which results in a forward–backward algorithm with double inertial effects and larger range of stepsize. We establish the linear convergence of the iterates to the unique solution using this algorithm. Furthermore, we provide convergence analysis for the class of strongly pseudo-monotone variational inequalities. We illustrate the effectiveness of our approach by applying it to structured optimization and pseudo-convex optimization problems.

In this paper, we suggest a new framework for analyzing primal subgradient methods for nonsmooth convex optimization problems. We show that the classical step-size rules, based on normalization of subgradient, or on knowledge of the optimal value of the objective function, need corrections when they are applied to optimization problems with constraints. Their proper modifications allow a significant acceleration of these schemes when the objective function has favorable properties (smoothness, strong convexity). We show how the new methods can be used for solving optimization problems with functional constraints with a possibility to approximate the optimal Lagrange multipliers. One of our primal-dual methods works also for unbounded feasible set.

Can order markets lead participants towards price-taking equilibrium? Viewing market sessions as steps of iterative algorithms, this paper indicates positive prospects for convergence. Mathematical arguments turn on convolution, efficiency and generalized gradients. Economic arguments revolve around reservation costs, derived from indifference or threshold payments for quantities supplied or demanded.

The notion of sharp minima, given by Polyak, is an important tool in studying the convergence analysis of algorithms designed to solve optimization problems. It has been studied extensively for variational inequality problems and equilibrium problems. In this paper, the convergence analysis of the sequence generated by proximal point method for quasi-equilibrium problem (QEP) will be established under sharp minima conditions. Further, the characterizations of weak sharp solution for QEP are provided. We also introduce an inexact proximal point method and demonstrate the convergence of the sequence for solving the QEP. Finally, we deduce the proximal point approximation for generalized Nash equilibrium problem.

In this paper, we first present necessary and sufficient conditions for the existence of global error bounds for a convex function without additional conditions on the function or the solution set. In particular, we obtain characterizations of such global error bounds in Euclidean spaces, which are often simple to check. Second, we prove that under a suitable assumption the subdifferential of the supremum function of an arbitrary family of convex continuous functions coincides with the convex hull of the subdifferentials of functions corresponding to the active indices at given points. As applications, we study the existence of global error bounds for infinite systems of linear and convex inequalities. Several examples are provided as well to explain the advantages of our results with existing ones in the literature.

This paper is concerned with recovering the solution of a final value problem associated with a parabolic equation involving a non linear source and a non-local term, which to the best of our knowledge has not been studied earlier. It is shown that the considered problem is ill-posed, and thus, some regularization method has to be employed in order to obtain stable approximations. In this regard, we obtain regularized approximations by solving some non linear integral equations which is derived by considering a truncated version of the Fourier expansion of the sought solution. Under different Gevrey smoothness assumptions on the exact solution, we provide parameter choice strategies and obtain the error estimates. A key tool in deriving such estimates is a version of Grönwalls’ inequality for iterated integrals, which perhaps, is proposed and analysed for the first time.

This paper studies constrained Markov decision processes under the total expected discounted cost optimality criterion, with a state-action dependent discount factor that may take any value between zero and one. Both the state and the action space are assumed to be Borel spaces. By using the linear programming approach, consisting in stating the control problem as a linear problem on a set of occupation measures, we show the existence of an optimal stationary Markov policy. Our results are based on the study of both weak-strong topologies in the space of occupation measures and Young measures in the space of Markov policies.