Journal of Optimization Theory and Applications最新文献

In this paper, we use the distributionally robust approach to study stochastic variational inequalities under the ambiguity of the true probability distribution, which is referred to as distributionally robust variational inequalities (DRVIs). First of all, we adopt a relaxed value function approach to relax the DRVI and obtain its relaxation counterpart. This is mainly motivated by the robust requirement in the modeling process as well as the possible calculation error in the numerical process. After that, we investigate qualitative convergence properties as the relaxation parameter tends to zero. Considering the perturbation of ambiguity sets, we further study the quantitative stability of the relaxation DRVI. Finally, when the ambiguity set is given by the general moment information, the discrete approximation of the relaxation DRVI is discussed.

In this paper we advocate for Isaacs’ method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin’s canonical optimal control example, which entails a double integrator plant. However, rather than controlling the state to the origin, we require the end state to reach a terminal set that contains the origin in its interior. Indeed, in practice, it is required to control to a prescribed tolerance rather than reach a desired end state; constraining the end state to a terminal manifold of co-dimension n − 1 renders the optimal control problem easier to solve. The global solution of the optimal control problem is obtained and the synthesized optimal control law is in state feedback form. In this respect, two target sets are considered: a smooth circular target and a square target with corners. Closed-loop state-feedback control laws are synthesized that drive the double integrator plant from an arbitrary initial state to the target set in minimum time. This is accomplished using Isaacs’ method for the solution of differential games, which entails dynamic programming (DP), working backward from the usable part of the target set, as opposed to obtaining the optimal trajectories using the necessary conditions for optimality provided by Pontryagin’s Maximum Principle (PMP). In this paper, the case is made for Isaacs’ method for the solution of differential games to be applied to the solution of optimal control problems by way of the juxtaposition of the PMP and DP methods.

In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.

We consider a simplicial branch and bound Global Optimization algorithm, where the search region is a simplex. Apart from using longest edge bisection, a simplicial partition set can be reduced due to monotonicity of the objective function. If there is a direction in which the objective function is monotone over a simplex, depending on whether the facets that may contain the minimum are at the border of the search region, we can remove the simplex completely, or reduce it to some of its border facets. Our research question deals with finding monotone directions and labeling facets of a simplex as border after longest edge bisection and reduction due to monotonicity. Experimental results are shown over a set of global optimization problems where the feasible set is defined as a simplex, and a global minimum point is located at a face of the simplicial feasible area.

The robust counterpart of a given conic linear program with uncertain data in the constraints is defined as the robust conic linear program that arises from replacing the nominal feasible set by the robust feasible set of points that remain feasible for any possible perturbation of the data within an uncertainty set. Any minor changes in the size of the uncertainty set can result in significant changes, for instance, in the robust feasible set, robust optimal value and the robust optimal set. The concept of quantifying the extent of these deviations is referred to as the stability of robustness. This paper establishes conditions for the stability of robustness under which minor changes in the size of the uncertainty sets lead to only minor changes in the robust feasible set of a given linear program with cone constraints and ball uncertainty sets.

In this paper, we consider a class of state constrained linear parabolic optimal control problems. Instead of treating the inequality state constraints directly, we reformulate the problem as an equality-constrained optimization problem, and then apply the augmented Lagrangian method (ALM) to solve it. We prove the convergence of the ALM without any existence or regularity assumptions on the corresponding Lagrange multipliers, which is an essential complement to the classical theoretical results for the ALM because restrictive regularity assumptions are usually required to guarantee the existence of the Lagrange multipliers associated with the state constraints. In addition, under an appropriate choice of penalty parameter sequence, we can obtain a super-linear non-ergodic convergence rate for the ALM. Computationally, we apply a semi-smooth Newton (SSN) method to solve the ALM subproblems and design an efficient preconditioned conjugate gradient method for solving the Newton systems. Some numerical results are given to illustrate the effectiveness and efficiency of our algorithm.

Multidimensional scaling (MDS) is a popular tool for dimensionality reduction and data visualization. Given distances between data points and a target low-dimension, the MDS problem seeks to find a configuration of these points in the low-dimensional space, such that the inter-point distances are preserved as well as possible. We focus on the most common approach to formulate the MDS problem, known as stress minimization, which results in a challenging non-smooth and non-convex optimization problem. In this paper, we propose an inertial version of the well-known SMACOF Algorithm, which we call AI-SMACOF. This algorithm is proven to be globally convergent, and to the best of our knowledge this is the first result of this kind for algorithms aiming at solving the stress MDS minimization. In addition to the theoretical findings, numerical experiments provide another evidence for the superiority of the proposed algorithm.

In this paper, we study a class of variational inequality problems the constraint set of which is the set of common solutions of a finite family of operator equations, involving hemi-continuous accretive operators on a reflexive and strictly convex Banach space with a Gâteaux differentiable norm. We present a sequential regularization method of Lavrentiev type and an iterative regularization one in combination with an inertial term to speed up convergence. The strong convergence of the methods is proved without the co-coercivity imposed on any operator in the family. An application of our results to solving the split common fixed point problem with pseudocontractive and nonexpansive operators is given with computational experiments for illustration.

A prototype model for a Fluid–Structure interaction is considered. We aim to stabilize [enhance stability of] the model by having access only to a portion of the state. Toward this goal we shall construct a compensator-based Luenberger design, with the following two goals: (1) reconstruct the original system asymptotically by tracking partial information about the full state, (2) stabilize the original unstable system by feeding an admissible control based on a system which is obtained from the compensator. The ultimate result is boundary control/stabilization of partially observed and originally unstable fluid–structure interaction with restricted information on the current state and without any knowledge of the initial condition. This prevents applicability of known methods in either open-loop or closed loop stabilization/control.