一类二层凸规划的对偶二层规划

本文讨论上层目标函数以下层子系统目标函数的最优值作为反馈的一类二层凸规划的对偶规划问题 ,在构成函数满足凸连续可微等条件的假设下 ,建立了二层凸规划的 Lagrange对偶二层规划 ,并证明了基本对偶定理 .     

集值优化问题的非线性增广拉格朗日方法

本文引入一类新的具有弱零极值性质的非线性增广罚函数,并利用增广拉格朗日方法和抽象共轭与双共轭,抽象次梯度,原问题稳定等概念来研究Banach空间中集值向量优化问题的非线性增广拉格朗日对偶定理·若原问题是稳定的,则原问题与对偶问题之间存在零对偶间隙。在下确界外稳定的假设下得到零对偶间隙性质成立的充分必要条件.这些结论是有限维空间上的实值优化问题和集值向量优化问题中相应结论的推广.     

基于一族蕴涵算子的对偶三I算法

三I算法是一种新的模糊推理方法,是传统的模糊推理方法的修改和补充. 三I表达式取最小值时的最优解算法(即对偶三I算法)是三I算法思想的延伸和完善.本文针对蕴涵算子族Ip,讨论了FMP和FMT问题的对偶三I算法,给出了相应的计算公式,从而也进一步促进了对三I算法的研究.     

一类线性与框式约束凸规划问题的原始-对偶内点算法

本文对一类具有线性和框式约束的凸规划问题给出了一个原始-对偶内点算法, 该算法可在任一原始-对偶可行内点启动, 并且全局收敛,当初始点靠近中心路径时, 算法成为中心路径跟踪算法。 数值实验表明, 算法对求解大型的这类问题是有效的。     

向量最优化问题的Lagrange对偶与择一定理

本文讨论无限维向量最优化问题的Lagrange对偶与弱对偶，建立了若干鞍点定理与弱鞍点定理．作为研究对偶问题的工具，建立了一个新的择一定理．     

锥约束复合优化问题的Lagrange对偶

利用共轭函数的上图性质,引入新的约束规范条件,等价刻画了带锥约束的复合优化问题与其Lagrange对偶问题之间的弱对偶,零对偶及强对偶,推广和改进了前人的相关结论.     

Orlicz对偶混合均质积分

该文讨论了Orlicz对偶混合均质积分的连续性、唯一性,给出了在一般线性变换下的性质,证明了关于Orlicz对偶混合均质积分的循环不等式,同时证明了关于Orlicz对偶混合均质积分的对偶Orlicz-Minkowski不等式与对偶均质积分关于调和Orlicz组合的对偶Orlicz-Brunn-Minkowski不等式是等价的,还得到了对偶Orlicz-Cauchy-Kubota公式.     

半无限规划的一个非线性Lagrangian对偶模型

本文给出半无限规划的一个对偶罚函数模型,该模型能处理目标函数不是凸函数的情形,从而凸(SIP)对偶为该模型的一个特例.并且,作为罚函数,本模型的罚因子比l1-罚函数要小,这使得算法更可行,最后,给出零对偶间隙证明.     

0-对偶预同态的一个充要条件

我们对带零逆半群定义了0-对偶预同态的概念,给出并证明了0-对偶预同态的一个等价条件.     

带约束广义变分不等式问题的一般分解算法的收敛性分析

本文考虑如下带约束广义变分不等式问题的增广Lagrangian对偶理论:寻找一点x∈Γ使满足,〈F(x),y-x〉 φ(x,y)-φ(x,x)≥0,y∈Γ,其中,Γ={y∈X｜Θ(y)∈-C}.对于求解这类一般变分不等式问题的基于增广Lagrangian对偶理论分解算法,本文给出了算法的收敛性分析.     

Nonlinear Lagrange Duality Theorems and Penalty Function Methods In Continuous Optimization

We propose a general dual program for a constrained optimization problem via generalized nonlinear Lagrangian functions. Our dual program includes a class of general dual programs with explicit structures as special cases. Duality theorems with the zero duality gap are proved under very general assumptions and several important corollaries which include some known results are given. Using dual functions as penalty functions, we also establish that a sequence of approximate optimal solutions of the penalty function converges to the optimal solution of the original optimization problem.     

Further Study on Augmented Lagrangian Duality Theory

In this paper, we present a necessary and sufficient condition for a zero duality gap between a primal optimization problem and its generalized augmented Lagrangian dual problems. The condition is mainly expressed in the form of the lower semicontinuity of a perturbation function at the origin. For a constrained optimization problem, a general equivalence is established for zero duality gap properties defined by a general nonlinear Lagrangian dual problem and a generalized augmented Lagrangian dual problem, respectively. For a constrained optimization problem with both equality and inequality constraints, we prove that first-order and second-order necessary optimality conditions of the augmented Lagrangian problems with a convex quadratic augmenting function converge to that of the original constrained program. For a mathematical program with only equality constraints, we show that the second-order necessary conditions of general augmented Lagrangian problems with a convex augmenting function converge to that of the original constrained program.This research is supported by the Research Grants Council of Hong Kong (PolyU B-Q359.)     

Nonlinear Lagrangian Theory for Nonconvex Optimization

The Lagrangian function in the conventional theory for solving constrained optimization problems is a linear combination of the cost and constraint functions. Typically, the optimality conditions based on linear Lagrangian theory are either necessary or sufficient, but not both unless the underlying cost and constraint functions are also convex.We propose a somewhat different approach for solving a nonconvex inequality constrained optimization problem based on a nonlinear Lagrangian function. This leads to optimality conditions which are both sufficient and necessary, without any convexity assumption. Subsequently, under appropriate assumptions, the optimality conditions derived from the new nonlinear Lagrangian approach are used to obtain an equivalent root-finding problem. By appropriately defining a dual optimization problem and an alternative dual problem, we show that zero duality gap will hold always regardless of convexity, contrary to the case of linear Lagrangian duality.     

Some Results about Duality and Exact Penalization

In this paper, we introduce the concept of the valley at 0 augmenting function and apply it to construct a class of valley at 0 augmented Lagrangian functions. We establish the existence of a path of optimal solutions generated by valley at 0 augmented Lagrangian problems and its convergence toward the optimal set of the original problem and obtain the zero duality gap property between the primal problem and the valley at 0 augmented Lagrangian dual problem. Moreover, we establish the exact penalization representation results in the framework of valley at 0 augmented Lagrangian.     

Second order optimality conditions for bilevel set optimization problems

In this paper, we introduce a new notion of augmenting function known as indicator augmenting function to establish a minmax type duality relation, existence of a path of solution converging to optimal value and a zero duality gap relation for a nonconvex primal problem and the corresponding Lagrangian dual problem. We also obtain necessary and sufficient conditions for an exact penalty representation in the framework of indicator augmented Lagrangian.     

On the sufficiency of finite support duals in semi-infinite linear programming

We consider semi-infinite linear programs with countably many constraints indexed by the natural numbers. When the constraint space is the vector space of all real valued sequences, we show that the finite support (Haar) dual is equivalent to the algebraic Lagrangian dual of the linear program. This settles a question left open by Anderson and Nash (1987). This result implies that if there is a duality gap between the primal linear program and its finite support dual, then this duality gap cannot be closed by considering the larger space of dual variables that define the algebraic Lagrangian dual. However, if the constraint space corresponds to certain subspaces of all real-valued sequences, there may be a strictly positive duality gap with the finite support dual, but a zero duality gap with the algebraic Lagrangian dual.     

Lagrangian methods for optimal control problems governed by a mixed quasi-variational inequality

This paper studies an optimal control problem where the state of the system is defined by a mixed quasi-variational inequality. Several sufficient conditions for the zero duality gap property between the optimal control problem and its nonlinear dual problem are obtained by using nonlinear Lagrangian methods. Our results are applied to an example where the mixed quasi-variational inequality leads to a bilateral obstacle problem.     

Exact augmented Lagrangian duality for mixed integer linear programming

We investigate the augmented Lagrangian dual (ALD) for mixed integer linear programming (MIP) problems. ALD modifies the classical Lagrangian dual by appending a nonlinear penalty function on the violation of the dualized constraints in order to reduce the duality gap. We first provide a primal characterization for ALD for MIPs and prove that ALD is able to asymptotically achieve zero duality gap when the weight on the penalty function is allowed to go to infinity. This provides an alternative characterization and proof of a recent result in Boland and Eberhard (Math Program 150(2):491–509, 2015, Proposition 3). We further show that, under some mild conditions, ALD using any norm as the augmenting function is able to close the duality gap of an MIP with a finite penalty coefficient. This generalizes the result in Boland and Eberhard (2015, Corollary 1) from pure integer programming problems with bounded feasible region to general MIPs. We also present an example where ALD with a quadratic augmenting function is not able to close the duality gap for any finite penalty coefficient.     

Unified Nonlinear Lagrangian Approach to Duality and Optimal Paths

In the context of an inequality constrained optimization problem, we present a unified nonlinear Lagrangian dual scheme and establish necessary and sufficient conditions for the zero duality gap property. From these results, we derive necessary and sufficient conditions for four classes of zero duality gap properties and establish the equivalence among them. Finally, we obtain the convergence of an optimal path for the unified scheme and present a sufficient condition for the finite termination of the optimal path. This research was partially supported by the Research Grants Council of Hong Kong Grant PolyU 5250/03E, the National Natural Science Foundation of China Grants 10471159 and 10571106, NCET, and the Natural Science Foundation of Chongqing     

On zero duality gap in nonconvex quadratic programming problems

We present in this paper new sufficient conditions for verifying zero duality gap in nonconvex quadratically/linearly constrained quadratic programs (QP). Based on saddle point condition and conic duality theorem, we first derive a sufficient condition for the zero duality gap between a quadratically constrained QP and its Lagrangian dual or SDP relaxation. We then use a distance measure to characterize the duality gap for nonconvex QP with linear constraints. We show that this distance can be computed via cell enumeration technique in discrete geometry. Finally, we revisit two sufficient optimality conditions in the literature for two classes of nonconvex QPs and show that these conditions actually imply zero duality gap.