Journal of Optimization Theory and Applications最新文献

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.

In this study, we propose an algorithmic framework for solving a class of optimal control problems. This class is associated with the minimum-time interception of moving target problems, where a plant with a given state equation must approach a moving target whose trajectory is known a priori. Our framework employs an analytical description of the distance from an arbitrary point to the reachable set of the plant. The proposed algorithm is always convergent and cannot be improved without losing the guarantee of its convergence to the correct solution for arbitrary Lipschitz continuous trajectories of the moving target. In practice, it is difficult to obtain an analytical description of the distance to the reachable set for the sophisticated state equation of the plant. Nevertheless, it was shown that the distance can be obtained for some widely used models, such as the Dubins car, in an explicit form. Finally, we illustrate the generality and effectiveness of the proposed framework for simple motions and the Dubins model.

This work introduces a new Riemannian optimization method for registering open parameterized surfaces with a constrained global optimization approach. The proposed formulation leads to a rigorous theoretic foundation and guarantees the existence and the uniqueness of a global solution. We also propose a new Bayesian clustering approach where local distributions of surfaces are modeled with spherical Gaussian processes. The maximization of the posterior density is performed with Hamiltonian dynamics which provide a natural and computationally efficient spherical Hamiltonian Monte Carlo sampling. Experimental results demonstrate the efficiency of the proposed method.

Many tasks in real life scenarios can be naturally formulated as nonconvex optimization problems. Unfortunately, to date, the iterative numerical methods to find even only the local minima of these nonconvex cost functions are extremely slow and strongly affected by the initialization chosen. We devise a predictor–corrector strategy that efficiently computes locally optimal solutions to these problems. An initialization-free convex minimization allows to predict a global good preliminary candidate, which is then corrected by solving a parameter-free nonconvex minimization. A simple algorithm, such as alternating direction method of multipliers works surprisingly well in producing good solutions. This strategy is applied to the challenging problem of decomposing a 1D signal into semantically distinct components mathematically identified by smooth, piecewise-constant, oscillatory structured and unstructured (noise) parts.