Weak minimizers,minimizers and variational inequalities for set-valued functions. A blooming wreath?

Recently, necessary and sufficient conditions in terms of variational inequalities have been introduced to characterize minimizers of convex set-valued functions. Similar results have been proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case, we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed, this might eventually prove the usefulness of the set optimization approach to renew the study of vector optimization.     

一种新的求解带约束的有限极大极小问题的精确罚函数

提出了一种新的精确光滑罚函数求解带约束的极大极小问题．仅仅添加一个额外的变量,利用这个精确光滑罚函数,将带约束的极大极小问题转化为无约束优化问题． 证明了在合理的假设条件下,当罚参数充分大,罚问题的极小值点就是原问题的极小值点．进一步,研究了局部精确性质．数值结果表明这种罚函数算法是求解带约束有限极大极小问题的一种有效算法．     

Robust penalty function method for an uncertain multi-time control optimization problems

This paper gives some new results on multi-time first-order PDE constrained control optimization problem in the face of data uncertainty (MCOPU). We obtain the robust sufficient optimality conditions for (MCOPU). Further, we construct an unconstrained multi-time control optimization problem (MCOPU)_? corresponding to (MCOPU) via absolute value penalty function method. Then, we show that the robust optimal solution to the constrained problem and a robust minimizer to the unconstrained problem are equivalent under suitable hypotheses. Moreover, we give some non-trivial examples to validate the results established in this paper.     

The cluster problem in constrained global optimization

Deterministic branch-and-bound algorithms for continuous global optimization often visit a large number of boxes in the neighborhood of a global minimizer, resulting in the so-called cluster problem (Du and Kearfott in J Glob Optim 5(3):253–265, 1994). This article extends previous analyses of the cluster problem in unconstrained global optimization (Du and Kearfott 1994; Wechsung et al. in J Glob Optim 58(3):429–438, 2014) to the constrained setting based on a recently-developed notion of convergence order for convex relaxation-based lower bounding schemes. It is shown that clustering can occur both on nearly-optimal and nearly-feasible regions in the vicinity of a global minimizer. In contrast to the case of unconstrained optimization, where at least second-order convergent schemes of relaxations are required to mitigate the cluster problem when the minimizer sits at a point of differentiability of the objective function, it is shown that first-order convergent lower bounding schemes for constrained problems may mitigate the cluster problem under certain conditions. Additionally, conditions under which second-order convergent lower bounding schemes are sufficient to mitigate the cluster problem around a global minimizer are developed. Conditions on the convergence order prefactor that are sufficient to altogether eliminate the cluster problem are also provided. This analysis reduces to previous analyses of the cluster problem for unconstrained optimization under suitable assumptions.     

A filled function method for constrained global optimization

In this paper, a filled function method for solving constrained global optimization problems is proposed. A filled function is proposed for escaping the current local minimizer of a constrained global optimization problem by combining the idea of filled function in unconstrained global optimization and the idea of penalty function in constrained optimization. Then a filled function method for obtaining a global minimizer or an approximate global minimizer of the constrained global optimization problem is presented. Some numerical results demonstrate the efficiency of this global optimization method for solving constrained global optimization problems.     

An Elastic Frictional Contact Problem with Unilateral Constraint

We consider a mathematical model which describes the equilibrium of an elastic body in contact with two obstacles. We derive its weak formulation which is in a form of an elliptic quasi-variational inequality for the displacement field. Then, under a smallness assumption, we establish the existence of a unique weak solution to the problem. We also study the dependence of the solution with respect to the data and prove a convergence result. Finally, we consider an optimization problem associated with the contact model for which we prove the existence of a minimizer and a convergence result, as well.     

A second-order dynamical system with Hessian-driven damping and penalty term associated to variational inequalities

Abstract

We consider the minimization of a convex objective function subject to the set of minima of another convex function, under the assumption that both functions are twice continuously differentiable. We approach this optimization problem from a continuous perspective by means of a second-order dynamical system with Hessian-driven damping and a penalty term corresponding to the constrained function. By constructing appropriate energy functionals, we prove weak convergence of the trajectories generated by this differential equation to a minimizer of the optimization problem as well as convergence for the objective function values along the trajectories. The performed investigations rely on Lyapunov analysis in combination with the continuous version of the Opial Lemma. In case the objective function is strongly convex, we can even show strong convergence of the trajectories.     

A Proximal-Type Method for Convex Vector Optimization Problem in Banach Spaces

In this paper, we consider a convex vector optimization problem of finding weak Pareto optimal solutions from a uniformly convex and uniformly smooth Banach space to a real Banach space, with respect to the partial order induced by a closed, convex, and pointed cone with a nonempty interior. We introduce a vector-valued proximal-type method based on the Lyapunov functional, carry out convergent analysis on this method, and prove that any sequence generated by the method weakly converges to a weak Pareto optimal solution of the vector optimization problem under some mild conditions. Our results improve and generalize some known results.     

An Exact Penalty Function Method for Continuous Inequality Constrained Optimal Control Problem

In this paper, we consider a class of optimal control problems subject to equality terminal state constraints and continuous state and control inequality constraints. By using the control parametrization technique and a time scaling transformation, the constrained optimal control problem is approximated by a sequence of optimal parameter selection problems with equality terminal state constraints and continuous state inequality constraints. Each of these constrained optimal parameter selection problems can be regarded as an optimization problem subject to equality constraints and continuous inequality constraints. On this basis, an exact penalty function method is used to devise a computational method to solve these optimization problems with equality constraints and continuous inequality constraints. The main idea is to augment the exact penalty function constructed from the equality constraints and continuous inequality constraints to the objective function, forming a new one. This gives rise to a sequence of unconstrained optimization problems. It is shown that, for sufficiently large penalty parameter value, any local minimizer of the unconstrained optimization problem is a local minimizer of the optimization problem with equality constraints and continuous inequality constraints. The convergent properties of the optimal parameter selection problems with equality constraints and continuous inequality constraints to the original optimal control problem are also discussed. For illustration, three examples are solved showing the effectiveness and applicability of the approach proposed.     

Vector duality for convex vector optimization problems by means of the quasi-interior of the ordering cone

We define weakly minimal elements of a set with respect to a convex cone by means of the quasi-interior of the cone and characterize them via linear scalarization, generalizing the classical weakly minimal elements from the literature. Then we attach to a general vector optimization problem, a dual vector optimization problem with respect to (generalized) weakly efficient solutions and establish new duality results. By considering particular cases of the primal vector optimization problem, we derive vector dual problems with respect to weakly efficient solutions for both constrained and unconstrained vector optimization problems and the corresponding weak, strong and converse duality statements.     

Characterization of vector strict global minimizers of order 2 in differentiable vector optimization problems under a new approximation method

In this paper, a new approximation method is introduced to characterize a so-called vector strict global minimizer of order 2 for a class of nonlinear differentiable multiobjective programming problems with (F,ρ)-convex functions of order 2. In this method, an equivalent vector optimization problem is constructed by a modification of both the objectives and the constraint functions in the original multiobjective programming problem at the given feasible point. In order to prove the equivalence between the original multiobjective programming problem and its associated F-approximated vector optimization problem, the suitable (F,ρ)-convexity of order 2 assumption is imposed on the functions constituting the considered vector optimization problem.     

关于集值优化问题的若干结果

引入了集值集值C-τ-半预不变凸概念，证明了集值集值C-τ-半预不变凸优化问题的局部弱有效元是弱有效元，给出了集值预不变凸变分不等式作为集值C-τ-半预不变凸优化问题的充分条件和必要条件，这些结果推广了文[1-4]的相应结果。     

Existence of Solutions to Time-Dependent Nonlinear Diffusion Equations via Convex Optimization

This paper aims at providing new existence results for time-dependent nonlinear diffusion equations by following a variational principle. More specifically, the nonlinear equation is reduced to a convex optimization problem via the Lagrange–Fenchel duality relations. We prove that, in the case when the potential related to the diffusivity function is continuous and has a polynomial growth with respect to the solution, the optimization problem is equivalent with the original diffusion equation. In the situation when the potential is singular, the minimization problem has a solution which can be viewed as a generalized solution to the diffusion equation. In this case, it is proved, however, that the null minimizer in the optimization problem in which the state boundedness is considered in addition is the weak solution to the original diffusion problem. This technique allows one to prove the existence in the cases when standard methods do not apply. The physical interpretation of the second case is intimately related to a flow in which two phases separated by a free boundary evolve in time, and has an immediate application to fluid filtration in porous media.     

总体优化一类双参数填充函数算法的改进

求解无约束总体优化问题的一类双参数填充函数算法需要假设该问题的局部极小解的个数只有有限个，而且填充函数中参数的选取与局部极小解的谷域的半径有关．该文对其填充函数作了适当改进，使得新的填充函数算法不仅无需对问题的局部极小解的个数作假设，而且填充函数中参数的选取与局部极小解的谷域的半径无关．数值试验表明算法是有效的．     

A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

一类不依赖于局部极小解个数的填充函数

求解无约束总体优化问题的一类单参数填充函数需要假设问题的局部极小解的个数只有有限个,而且填充函数中参数的选取与局部极小解的谷域的半径有关.本文对填充函数的定义作适当改进,而且对已有的这一类填充函数作改进,构造了一类双参数填充函数.新的填充函数不仅无须对问题的局部极小解的个数作假设,而且其中参数的选取与局部极小解的谷域的半径无关.     

无约束优化问题的一个下降方法

本文研究了无约束优化问题.利用当前和前面迭代点的信息以及曲线搜索技巧产生新的迭代点,得到了一个新的求解无约束优化问题的下降方法.在较弱条件下证明了算法具有全局收敛性.当目标函数为一致凸函数时,证明了算法具有线性收敛速率.初步的数值试验表明算法是有效的.     

A Proximal Point-Type Method for Multicriteria Optimization

In this paper, we present a proximal point algorithm for multicriteria optimization, by assuming an iterative process which uses a variable scalarization function. With respect to the convergence analysis, firstly we show that, for any sequence generated from our algorithm, each accumulation point is a Pareto critical point for the multiobjective function. A more significant novelty here is that our paper gets full convergence for quasi-convex functions. In the convex or pseudo-convex cases, we prove convergence to a weak Pareto optimal point. Another contribution is to consider a variant of our algorithm, obtaining the iterative step through an unconstrained subproblem. Then, we show that any sequence generated by this new algorithm attains a Pareto optimal point after a finite number of iterations under the assumption that the weak Pareto optimal set is weak sharp for the multiobjective problem.     

Constrained and unconstrained rearrangement minimization problems related to the <Emphasis Type="Italic">p</Emphasis>-Laplace operator

In this paper we consider an unconstrained and a constrained minimization problem related to the boundary value problem

$$ - {\Delta _p}u = f{\text{ in }}D,{\text{ }}u = 0{\text{ on }}\partial D$$

. In the unconstrained problem we minimize an energy functional relative to a rearrangement class, and prove existence of a unique solution. We also consider the case when D is a planar disk and show that the minimizer is radial and increasing. In the constrained problem we minimize the energy functional relative to the intersection of a rearrangement class with an affine subspace of codimension one in an appropriate function space. We briefly discuss our motivation for studying the constrained minimization problem.