Are dualities appropriate for duality theories in optimization?

We raise some questions about duality theories in global optimization. The main one concerns the possibility to extend the use of conjugacies to general dualities for studying dual optimization problems. In fact, we examine whether dualities are the most general concepts to get duality results. We also consider the passage from a Lagrangian approach to a perturbational approach and the reverse passage in the framework of general dualities. Since a notion of subdifferential can be defined for any duality, it is natural to examine whether the familiar interpretation of multipliers as generalized derivatives of the performance function associated with a dualizing parameterization of the given problem is still valid in the general framework of dualities.     

有限位移理论线弹性力学二类和三类混合变量的变分原理及其应用

提出了有限位移理论线弹性力学二类混合变量和三类混合变量的变分原理.考虑已知边界条件的变化并应用有限位移理论的功的互等定理,在导出上述两类变分原理的过程中起到了关键作用和桥梁作用.首先,考虑已知位移边界条件的变化和应用功的互等定理,导出了二类混合变量的最小势能原理.用类似的方法,导出了二类混合变量的驻值余能原理.应用应变能密度和应力余能密度的关系式于上述两个变分原理,得到三类混合变量的变分原理.然后,给出了二类和三类混合变量的虚功原理和虚余功原理.同时,应用拉氏乘子法导出了广义变分原理.以一个算例说明了在某些情况下拉氏乘子法会失效,介绍了构成广义变分原理泛函的半逆法.最后,应用二类混合变量最小势能原理计算了一大挠度悬臂梁的弯曲.     

粘性流体力学的变分原理和广义变分原理

本文建立了不可压缩和可压缩粘性流体力学问题的变分原理,即最大功率消耗原理和它们的广义变分原理.     

Deriving generalized variational principles in general mechanics by using Lagrangian multiplier method

By using the involutory transformations, the classical variational principle—Hamiltonian principle— of two kinds of variables in general mechanics is advanced and by using undetermined Lagrangian multiplier method, the generalized variational principles and generalized variational principles with subsidiary conditions are established. The stationary conditions of various kinds of variational principles are derived and the relational problems discussed. Project supported by the National Natural Science Foundation of China (Grant No. 19872022) and the Doctoral Education Foundation of China (Grant No. 97021710).     

大应变固结理论的分区变分原理及其广义变分原理

土体材料本构特性的差异问题与大变形问题是分析岩土材料变形特性的基本问题．根据有限变形的描述方法构筑土体结构大变形固结方程,证明了大变形固结的变分原理A·D2应用分区子结构的连续条件,推导固结理论的分区变分原理．引用Lagrange乘子法构筑并证明了大变形固结问题在无约束状态下的广义分区变分原理．     

Variational Dualities in the Linear Regulator and Estimation Problems With and Without Time Delay

A unified approach to the variational dualities in the stateregulator of control theory and linear estimation with and withouttime delay is developed by the complementary variational technique.New variational duals for the state regulator with time-delayand estimation problems with and without time delay are presented.     

Reduced Variational Formulations in Free Boundary Continuum Mechanics

We present the material, spatial, and convective representations for elasticity and fluids with a free boundary from the Lagrangian reduction point of view, using the material and spatial symmetries of these systems. The associated constrained variational principles are formulated and the resulting equations of motion are deduced. In addition, we introduce general free boundary continua that contain both elasticity and free boundary hydrodynamics, extend for them various classical notions, and present the constrained variational principles and the equations of motion in the three representations.     

The construction of dual variational principles of the theory of the seepage of an incompressible fluid in some deformable complex media

A method of constructing dual variational principles for problems of the seepage of an incompressible fluid in deformable media with complex rheology is presented. Dual variational principles of the fluid seepage in a typical deformable media with complex rheology [1, 2], namely, a Maxwell element, connected in parallel with a visco-plastic or elastic element, are constructed. The variational principles are derived from variational problems compiled for the constitutive relations for fluid and solid phases.     

Dual variational principles with deviating argument and discontinuous extremals

We study dual or complementary variational principles for functionals with deviating argument and discontinuous extremals. Local conditions for the existence of a pair of dual extremum principles are given. Results are then applied to the control problem involving linear neutral differential-difference equations.     

Macro-Hybrid Variational Formulations of Constrained Boundary Value Problems

In the context of convex analysis, macro-hybrid variational formulations of constrained boundary value problems are presented. Monotone mixed variational inclusions are macro-hybridized on the basis of nonoverlapping domain decompositions, and corresponding three-field versions are derived. Then, for regularization purposes, augmented formulations are established via preconditioned exact penalizations and expressed in terms of proximation operators. Optimization interpretations are given for potential problems, recovering the classic two- and three-field augmented Lagrangian formulations. Furthermore, associated parallel two- and three-field proximal-point algorithms are discussed for numerical resolution of finite element discretizations. Applications to dual mixed variational formulations of problems from mechanics illustrate the theory.     

A method of duality for a mixed vector equilibrium problem

In this paper, a dual scheme for a mixed vector equilibrium problem is introduced by using the method of Fenchel conjugate function. Under the stabilization condition, the relationships between the solutions of mixed vector equilibrium problem (MVEP) and dual mixed vector equilibrium problem (DMVEP) are discussed. Moreover, under the same condition, the solutions of MVEP and DMVEP are proved relating to the saddle points of an associated Lagrangian mapping. As applications, this dual scheme is applied to vector convex optimization and vector variational inequality.     

非完整非有势系统相对于非惯性系的广义Noether定理

本文构造了力学系统相对运动的Lagrange函数,建立了非线性非完整非有势系统相对于非惯性系的Jourdain型变分原理,提出并证明了这类力学系统相对于非惯性系的广义Noether定理,研究了其守恒量.     

Natural dualities for three classes of relational structures

Motivated by the work of Hofmann [14] and Davey [7] on dualities for structures, we will construct natural dualities for the classes of quasi-ordered sets, equivalence-relationed sets, and reflexive (undirected) graphs. We describe the dual structures by giving finite sets of quasi-equations that axiomatise the dual categories. We also show that there is a natural connection between the duality we construct for finite quasi-ordered sets and Birkhoff’s representation theorem for finite ordered sets [2], and between the dualities constructed for quasi-ordered sets and equivalence-relationed sets.     

Duality for generalized equilibrium problem

We introduce a generalized equilibrium problem (GEP) that allow us to develop a robust dual scheme for this problem, based on the theory of conjugate functions. We obtain a unified dual analysis for interesting problems. Indeed, the Lagrangian duality for convex optimization is a particular case of our dual problem. We establish necessary and sufficient optimality conditions for GEP that become a well-known theorem given by Mosco and the dual results obtained by Morgan and Romaniello, which extend those introduced by Auslender and Teboulle for a variational inequality problem.     

A dual form of Noether's theorem with applications to continuum mechanics

Considering simultaneously the equations of motion of the physical system and of the non-physical adjoint system, we introduce a general form of Noether's theorem by constructing a “dual Lagrangian” functional with a corresponding invariant of motion which preserves its value along the trajectories of combined physical and unphysical systems. The statement of invariance of this functional reduces to the classical statement of Noether's theorem if the system is self-adjoint; some possible generalizations are indicated. Applications to continuum mechanics are discussed within the framework of Noble's dual variational formulation.     

Lagrangian Approach to Quasiconvex Programing

For a mathematical programming problem, we consider a Lagrangian approach inspired by quasiconvex duality, but as close as possible to the usual convex Lagrangian. We focus our attention on the set of multipliers and we look for their interpretation as generalized derivatives of the performance function associated with a simple perturbation of the given problem. We do not use quasiconvex dualities, but simple direct arguments.     

Composition Duality Methods for Mixed Variational Inclusions

Composition duality methods for mixed variational inclusions are studied in a functional framework of reflexive Banach spaces. On the basis of duality principles, the solvability of maximal monotone and subdifferential mixed variational inclusions is established. For computational purposes, mass-preconditioned augmented formulations are introduced for regularization, as well as three-field and macro-hybrid variational versions. At a finite-dimensional level, corresponding discrete mixed and macro-hybrid internal approximations are discussed, as well as proximal-point iterative algorithms. Primal and dual mixed variational inclusions from contact mechanics illustrate the theory.     

Tangential extremal principles for finite and infinite systems of sets, I: basic theory

In this paper we develop new extremal principles in variational analysis that deal with finite and infinite systems of convex and nonconvex sets. The results obtained, unified under the name of tangential extremal principles, combine primal and dual approaches to the study of variational systems being in fact first extremal principles applied to infinite systems of sets. The first part of the paper concerns the basic theory of tangential extremal principles while the second part presents applications to problems of semi-infinite programming and multiobjective optimization.     

Dual extremum principles in finite deformation elastoplastic analysis

Dual extremum principles are established in this paper for the variational boundary-value problem of elasto-perfect plasticity with large deformation. There exists a duality gap between the primal and dual variational problems. Our application to nonlinear limit analysis yields a pair of dual bounding theorems for the safety factor, when the gap has the right sign. It is proved that both the upper and lower bounds directly depend on the properties of the dual gap function.This research was supported by Army Research Office grant DAAL 03-86-K 0171 and National Science Foundation grant DMS-87-03313.     

Augmented Lagrangian method for second-order cone programs under second-order sufficiency

This paper addresses problems of second-order cone programming important in optimization theory and applications. The main attention is paid to the augmented Lagrangian method (ALM) for such problems considered in both exact and inexact forms. Using generalized differential tools of second-order variational analysis, we formulate the corresponding version of second-order sufficiency and use it to establish, among other results, the uniform second-order growth condition for the augmented Lagrangian. The latter allows us to justify the solvability of subproblems in the ALM and to prove the linear primal–dual convergence of this method.