A Volume Constrained Variational Problem with Lower-Order Terms

We study a one-dimensional variational problem with two or more level set constraints. The existence of global and local minimizers turns out to be dependent on the regularity of the energy density. A complete characterization of local minimizers and the underlying energy landscape is provided. The Γ -limit when the phases exhaust the whole domain is computed.     

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

From a Nonlinear, Nonconvex Variational Problem to a Linear, Convex Formulation

   Abstract. We propose a general approach to deal with nonlinear, nonconvex variational problems based on a reformulation of the problem resulting in an optimization problem with linear cost functional and convex constraints. As a first step we explicitly explore these ideas to some one-dimensional variational problems and obtain specific conclusions of an analytical and numerical nature.     

Optimal Control of Elliptic Variational Inequalities

Optimality systems for optimal control problems governed by elliptic variational inequalities are derived. Existence of appropriately defined Lagrange multipliers is proved. A primal—dual active set method is proposed to solve the optimality systems numerically. Examples with and without lack of strict complementarity are included. Accepted 5 March 1999     

A Simply Constrained Optimization Reformulation of KKT Systems Arising from Variational Inequalities

The Karush—Kuhn—Tucker (KKT) conditions can be regarded as optimality conditions for both variational inequalities and constrained optimization problems. In order to overcome some drawbacks of recently proposed reformulations of KKT systems, we propose casting KKT systems as a minimization problem with nonnegativity constraints on some of the variables. We prove that, under fairly mild assumptions, every stationary point of this constrained minimization problem is a solution of the KKT conditions. Based on this reformulation, a new algorithm for the solution of the KKT conditions is suggested and shown to have some strong global and local convergence properties. Accepted 10 December 1997     

Explicit Solutions of Nonconvex Variational Problems in Dimension One

We propose a method of finding the generalized solutions of nonconvex variational problems by solving an appropriate differential inclusion that is motivated by necessary conditions of optimality for such generalized minimizers. Accepted 28 September 1998     

A Relaxation of a Min-Cut Problem in an Anisotropic Continuous Network

Strang [18] introduced optimization problems on a Euclidean domain which are closely related with problems in mechanics and noted that the problems are regarded as continuous versions of famous max-flow and min-cut problems. In [15] we generalized the problems and called the generalized problems max-flow and min-cut problems of Strang's type. In this paper we formulate a relaxed version of the min-cut problem of Strang's type and prove the existence of optimal solutions under some suitable conditions. The conditions are essential. In fact, there is an example of the relaxed version which has no optimal solutions if the conditions are not fulfilled. We give such an example in the final section. Accepted 8 October 1998     

A new equilibrium in the one-sided asymmetric information market with pairwise meetings

We reconsider the model used by Serrano and Yosha (1993) who were interested in information revelation in markets with pairwise meetings. We prove that there exists an additional equilibrium not detected in the original paper and show that this equilibrium is characterized by incomplete revelation of information which was not the case of the other already identified equilibria of the model.     

On a Problem of Daubechies

   Abstract. We solve a problem posed by Daubechies [12] by showing the nonexistence of orthonormal wavelet bases with good time-frequency localization associated with irrational dilation factors.     

A Modified Alternating Direction Method for Variational Inequality Problems

The alternating direction method is an attractive method for solving large-scale variational inequality problems whenever the subproblems can be solved efficiently. However, the subproblems are still variational inequality problems, which are as structurally difficult to solve as the original one. To overcome this disadvantage, in this paper we propose a new alternating direction method for solving a class of nonlinear monotone variational inequality problems. In each iteration the method just makes an orthogonal projection to a simple set and some function evaluations. We report some preliminary computational results to illustrate the efficiency of the method. Accepted 4 May 2001. Online publication 19 October, 2001.     

Variational Inequalities for Leavable Bounded-Velocity Control

We study the variational inequality associated with a bounded-velocity control problem when discretionary stopping is allowed. We establish the existence of a strong solution by using the viscosity solution techniques. The optimal policy is shown to exist from the optimality conditions in the variational inequality.     

Variational Inequalities for Leavable Bounded-Velocity Control

We study the variational inequality associated with a bounded-velocity control problem when discretionary stopping is allowed. We establish the existence of a strong solution by using the viscosity solution techniques. The optimal policy is shown to exist from the optimality conditions in the variational inequality.     

A one-dimensional variational problem with gradient constraint

In this note we give an explicit characterization of the minimum value of one-dimensional integral variational problem with gradient constraint by a positive measurable function.     

有约束的变测度积分——水平集算法

提出了一种有约束的变测度积分-水平集的算法,对不同的箱子采用不同的测度,结合确定性数论方法选取一致分布佳点集来代替Monte-Carlo随机投点,使水平值充分地下降,更快地到达全局最小,从而提高算法的计算效率.给出了算法的收敛性证明,并通过数值算例验证了它的有效性.     

线性约束最优化的一个共轭投影梯度法

本结合共轭梯度法及梯度投影法的思想,建立线性等式约束最优化的一个新算法,称之为共轭投影梯度法。分别对二次凸目标函数和一般目标函数分析和论证了算法的重要性质和收敛性。     

Asymptotic Behavior of a Geman and McClure Discrete Model

In this paper we consider a class of discrete variational models derived from a theory of Geman and McClure (see [15]) and study their asymptotic behavior when their stepsize tends to zero. It is shown that a result of Γ -convergence toward a certain functional holds true if a characteristic parameter of these models obeys a well-defined dependence law upon the stepsize. Under this condition the Γ -limit is a modified form of the Mumford—Shah functional. Accepted 19 June 1998     

Lipschitzian Regularity of Minimizers for Optimal Control Problems with Control-Affine Dynamics

We study the Lagrange Problem of Optimal Control with a functional and control-affine dynamics = f(t,x) + g(t,x)u and (a priori) unconstrained control u∈ \bf R ^m . We obtain conditions under which the minimizing controls of the problem are bounded—a fact which is crucial for the applicability of many necessary optimality conditions, like, for example, the Pontryagin Maximum Principle. As a corollary we obtain conditions for the Lipschitzian regularity of minimizers of the Basic Problem of the Calculus of Variations and of the Problem of the Calculus of Variations with higher-order derivatives. Accepted 15 March 1999     

解非线性约束优化问题的依赖域算法

本文对一般非线性约束优化问题提出了一个信赖域算法,导出了等价的KKT条件.在试探步满足适当条件下,证明了算法的全局收敛性,并进行了数值试验.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.