Progressive genetic algorithm for solution of optimization problems with nonlinear equality and inequality constraints

A new approach, identified as progressive genetic algorithm (PGA), is proposed for the solutions of optimization problems with nonlinear equality and inequality constraints. Based on genetic algorithms (GAs) and iteration method, PGA divides the optimization process into two steps; iteration and search steps. In the iteration step, the constraints of the original problem are linearized using truncated Taylor series expansion, yielding an approximate problem with linearized constraints. In the search step, GA is applied to the problem with linearized constraints for the local optimal solution. The final solution is obtained from a progressive iterative process. Application of the proposed method to two simple examples is given to demonstrate the algorithm.     

On certain optimization methods with finite-step inner algorithms for convex finite-dimensional problems with inequality constraints

Numerical methods are proposed for solving finite-dimensional convex problems with inequality constraints satisfying the Slater condition. A method based on solving the dual to the original regularized problem is proposed and justified for problems having a strictly uniformly convex sum of the objective function and the constraint functions. Conditions for the convergence of this method are derived, and convergence rate estimates are obtained for convergence with respect to the functional, convergence with respect to the argument to the set of optimizers, and convergence to the g-normal solution. For more general convex finite-dimensional minimization problems with inequality constraints, two methods with finite-step inner algorithms are proposed. The methods are based on the projected gradient and conditional gradient algorithms. The paper is focused on finite-dimensional problems obtained by approximating infinite-dimensional problems, in particular, optimal control problems for systems with lumped or distributed parameters.     

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.)     

A finite steps algorithm for solving convex feasibility problems

This paper develops a new variant of the classical alternating projection method for solving convex feasibility problems where the constraints are given by the intersection of two convex cones in a Hilbert space. An extension to the feasibility problem for the intersection of two convex sets is presented as well. It is shown that one can solve such problems in a finite number of steps and an explicit upper bound for the required number of steps is obtained. As an application, we propose a new finite steps algorithm for linear programming with linear matrix inequality constraints. This solution is computed by solving a sequence of a matrix eigenvalue decompositions. Moreover, the proposed procedure takes advantage of the structure of the problem. In particular, it is well adapted for problems with several small size constraints.     

Local Linear Convergence of an Outer Approximation Projection Method for Variational Inequalities

This paper considers an outer approximation projection method for variational inequalities, in which the projections are not performed on the original set that appears in the variational inequality, but on a polyhedral convex set defined by the linearized constraints. It shows that the method converges linearly, when the starting point is sufficiently close to the solution and the step lengths are sufficiently small.     

Duality for multiobjective optimization problems with convex objective functions and D.C. constraints

In this paper we provide a duality theory for multiobjective optimization problems with convex objective functions and finitely many D.C. constraints. In order to do this, we study first the duality for a scalar convex optimization problem with inequality constraints defined by extended real-valued convex functions. For a family of multiobjective problems associated to the initial one we determine then, by means of the scalar duality results, their multiobjective dual problems. Finally, we consider as a special case the duality for the convex multiobjective optimization problem with convex constraints.     

Second-order global optimality conditions for convex composite optimization

In recent years second-order sufficient conditions of an isolated local minimizer for convex composite optimization problems have been established. In this paper, second-order optimality conditions are obtained of aglobal minimizer for convex composite problems with a non-finite valued convex function and a twice strictly differentiable function by introducing a generalized representation condition. This result is applied to a minimization problem with a closed convex set constraint which is shown to satisfy the basic constraint qualification. In particular, second-order necessary and sufficient conditions of a solution for a variational inequality problem with convex composite inequality constraints are obtained. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.     

Averaged Subgradient Methods for Constrained Convex Optimization and Nash Equilibria Computation

We consider the problem of finding minima of convex functions under convex inequality constraints as well as the problem of finding Nash equilibria in n -person constant sum games. We prove that both problems can be solved by algorithms whose basic principles consist of representing the original problems as infinite systems of convex inequalities which, in turn, can be approached by outer projection techniques. Experiments showing how one of these algorithms behaves in test cases are presented and, in context, we describe a numerical method for computing subgradients of convex functions.     

ε-Pareto Optimality Conditions for Convex Multiobjective Programming via Max Function

We consider a nondifferentiable convex multiobjective optimization problem whose feasible set is defined by affine equality constraints, convex inequality constraints, and an abstract convex set constraint. We obtain Fritz John and Kuhn–Tucker necessary and sufficient conditions for ε-Pareto optimality via a max function. We also provide some relations among ε-Pareto solutions for such a problem and approximate solutions for several associated scalar problems.     

Successive Optimization Method via Parametric Monotone Composition Formulation*

In this paper a successive optimization method for solving inequality constrained optimization problems is introduced via a parametric monotone composition reformulation. The global optimal value of the original constrained optimization problem is shown to be the least root of the optimal value function of an auxiliary parametric optimization problem, thus can be found via a bisection method. The parametric optimization subproblem is formulated in such a way that it is a one-parameter problem and its value function is a monotone composition function with respect to the original objective function and the constraints. Various forms can be taken in the parametric optimization problem in accordance with a special structure of the original optimization problem, and in some cases, the parametric optimization problems are convex composite ones. Finally, the parametric monotone composite reformulation is applied to study local optimality.     

三个可分离算子凸优化的线性化方法

本文研究了三个可分离算子不含交叉变量的线性约束凸优化问题.利用定制的邻近点算法,对其变分不等式子问题进行线性化处理,并增加一邻近点项,使其子问题成为易于运算的单调线性变分不等式,得到了线性化定制的邻近点算法,并证明了全局收敛性,推广了文献中的研究结果.     

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.     

Gap Function Approach to the Generalized Nash Equilibrium Problem

We consider an optimization reformulation approach for the generalized Nash equilibrium problem (GNEP) that uses the regularized gap function of a quasi-variational inequality (QVI). The regularized gap function for QVI is in general not differentiable, but only directionally differentiable. Moreover, a simple condition has yet to be established, under which any stationary point of the regularized gap function solves the QVI. We tackle these issues for the GNEP in which the shared constraints are given by linear equalities, while the individual constraints are given by convex inequalities. First, we formulate the minimization problem involving the regularized gap function and show the equivalence to GNEP. Next, we establish the differentiability of the regularized gap function and show that any stationary point of the minimization problem solves the original GNEP under some suitable assumptions. Then, by using a barrier technique, we propose an algorithm that sequentially solves minimization problems obtained from GNEPs with the shared equality constraints only. Further, we discuss the case of shared inequality constraints and present an algorithm that utilizes the transformation of the inequality constraints to equality constraints by means of slack variables. We present some results of numerical experiments to illustrate the proposed approach.     

First and Second Order Necessary Conditions for Stochastic Optimal Control Problems

In this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state.     

A primal-dual interior point method for optimal zero-forcing beamformer design under per-antenna power constraints

In this paper, we consider an optimal zero-forcing beamformer design problem in multi-user multiple-input multiple-output broadcast channel. The minimum user rate is maximized subject to zero-forcing constraints and power constraint on each base station antenna array element. The natural formulation leads to a nonconvex optimization problem. This problem is shown to be equivalent to a convex optimization problem with linear objective function, linear equality and inequality constraints and quadratic inequality constraints. Here, the indirect elimination method is applied to reduce the convex optimization problem into an equivalent convex optimization problem of lower dimension with only inequality constraints. The primal-dual interior point method is utilized to develop an effective algorithm (in terms of computational efficiency) via solving the modified KKT equations with Newton method. Numerical simulations are carried out. Compared to algorithms based on a trust region interior point method and sequential quadratic programming method, it is observed that the method proposed is much superior in terms of computational efficiency.     

Exact penalty functions method for mathematical programming problems involving invex functions

In this paper, some new results on the exact penalty function method are presented. Simple optimality characterizations are given for the differentiable nonconvex optimization problems with both inequality and equality constraints via exact penalty function method. The equivalence between sets of optimal solutions in the original mathematical programming problem and its associated exact penalized optimization problem is established under suitable invexity assumption. Furthermore, the equivalence between a saddle point in the invex mathematical programming problem and an optimal point in its exact penalized optimization problem is also proved.     

ε-Optimal solutions in nondifferentiable convex programming and some related questions

In this paper we present ε-optimality conditions of the Kuhn-Tucker type for points which are within ε of being optimal to the problem of minimizing a nondifferentiable convex objective function subject to nondifferentiable convex inequality constraints, linear equality constraints and abstract constraints. Such ε-optimality conditions are of interest for theoretical consideration as well as from the computational point of view. Some illustrative applications are made. Thus we derive an expression for the ε-subdifferential of a general convex ‘max function’. We also show how the ε-optimality conditions given in this paper can be mechanized into a bundle algorithm for solving nondifferentiable convex programming problems with linear inequality constraints.     

求解带线性约束的凸优化的一类自适应不定线性化增广拉格朗日方法

增广拉格朗日方法是求解带线性约束的凸优化问题的有效算法.线性化增广拉格朗日方法通过线性化增广拉格朗日函数的二次罚项并加上一个临近正则项,使得子问题容易求解,其中正则项系数的恰当选取对算法的收敛性和收敛速度至关重要.较大的系数可保证算法收敛性,但容易导致小步长.较小的系数允许迭代步长增大,但容易导致算法不收敛.本文考虑求解带线性等式或不等式约束的凸优化问题.我们利用自适应技术设计了一类不定线性化增广拉格朗日方法,即利用当前迭代点的信息自适应选取合适的正则项系数,在保证收敛性的前提下尽量使得子问题步长选择范围更大,从而提高算法收敛速度.我们从理论上证明了算法的全局收敛性,并利用数值实验说明了算法的有效性.     

求解约束极大极小问题的一种熵函数法

１引言熵函数法的原始思想源于Ｋｒｅｉｓｓｅｌｍｅｉｅｒ和Ｓｔｅｉｎｈａｕｓｅｒ于１９７９年发表的文［１］．由于使用该方法容易编制可以求解多类优化问题的通用软件,并在具有某种凸性的情况下都能求得满足工程精度要求的解,因而受到国内外工程技术人员的喜爱,进入八十年代以来,该方法被广泛地应用于结构优化和工程设计等领域［２－５］．近年来,熵函数法在求解约束和无约束极大极小问题、线性规划以及半无限规划等问题的算法研究中,也取得了一些很好的成果［６－９］带有等式或不等式约束的极大极小问题是一类具有广泛代表性的…     

Approximation and optimization of discrete and differential inclusions described by inequality constraints

In the first part of this article optimization of polyhedral discrete and differential inclusions is considered, the problem is reduced to convex minimization problem and the necessary and sufficient condition for optimality is derived. The optimality conditions for polyhedral differential inclusions based on discrete-approximation problem according to continuous problems are formulated. In particular, boundedness of the set of adjoint discrete solutions and upper semi-continuity of the locally adjoint mapping are proved. In the second part of this article an optimization problem described by convex inequality constraint is studied. By using the equivalence theorem concerning the subdifferential calculus and approximating method necessary and sufficient condition for discrete-approximation problem with inequality constraint is established.