Parametric dual regularization for an optimal control problem with pointwise state constraints

The perturbation method is used in the dual regularization theory for a linear convex optimal control problem with a strongly convex objective functional and pointwise state constraints understood as ones in L ₂. Primary attention is given to the qualitative properties of the dual regularization method, depending on the differential properties of the value function (S-function) in the optimization problem. It is shown that the convergence of the method is closely related to the Lagrange principle and the Pontryagin maximum principle. The dual regularization scheme is shown to provide a new method for proving the maximum principle in the problem with pointwise state constraints understood in L ₂ or C. The regularized Lagrange principle in nondifferential form and the regularized Pontryagin maximum principle are discussed. Illustrative examples are presented.     

Kuhn-Tucker定理的精确形式

本文对有序向量空间中凸规划问题的Kuhn－Tucker定理提出一个精确形式，即将Lagrange因子存在性转化为一个新问题最优值的有界性．     

Sequential stable Kuhn-Tucker theorem in nonlinear programming

A general parametric nonlinear mathematical programming problem with an operator equality constraint and a finite number of functional inequality constraints is considered in a Hilbert space. Elements of a minimizing sequence for this problem are formally constructed from elements of minimizing sequences for its augmented Lagrangian with values of dual variables chosen by applying the Tikhonov stabilization method in the course of solving the corresponding modified dual problem. A sequential Kuhn-Tucker theorem in nondifferential form is proved in terms of minimizing sequences and augmented Lagrangians. The theorem is stable with respect to errors in the initial data and provides a necessary and sufficient condition on the elements of a minimizing sequence. It is shown that the structure of the augmented Lagrangian is a direct consequence of the generalized differentiability properties of the value function in the problem. The proof is based on a “nonlinear” version of the dual regularization method, which is substantiated in this paper. An example is given illustrating that the formal construction of a minimizing sequence is unstable without regularizing the solution of the modified dual problem.     

A generalization of the Gronwall-Bellman lemma and its applications

In this study we present an important theorem of the alternative involving convex functions and convex cones. From this theorem we develop saddle value optimality criteria and stationary optimality criteria for convex programs. Under suitable constraint qualification we obtain a generalized form of the Kuhn-Tucker conditions. We also use the theorem of the alternative in developing an important duality theorem. No duality gaps are encountered under the constraint qualification imposed earlier and the dual problem always possesses a solution. Moreover, it is shown that all constraint qualifications assure that the primal problem is stable in the sense used by Gale and others. The notion of stability is closely tied up with the positivity of the lagrangian multiplier of the objective function.     

拟不变凸集值优化问题严有效解的最优性条件

在赋范线性空间中借助切导数研究集值优化问题的严有效性．当目标函数和约束函数相对于同一向量函数为拟不变凸时,利用凸集分离定理给出了集值优化问题取得严有效元的Kuhn—Xhcker型最优陛必要条件．利用切导数的性质,用构造性方法得到了拟不变凸集值优化问题取得严有效元的充分条件．     

Convex two-level optimization problem

A two-level optimization problem is considered in which the objective functional of the second-level problem is minimized on the solution set of the first-level problem. Convergence of the modified penalty method is established. The main results include a continuous two-level optimization method based on the regularized extremal shifting principle [3, 5, 6]. For a linearly convex problem, two-sided bounds on the approximation by the first-level functional are established in addition to convergence. For a linearly quadratic problem, two-sided bounds on the approximation by the second-level functional are derived. For a linearly quadratic problem with interval constraints, an explicit form of differential inclusions is presented for the implementation of the method. __________ Translated from Nelineinaya Dinamika i Upravlenie, No. 4, pp. 257–286, 2004.     

Stable sequential convex programming in a Hilbert space and its application for solving unstable problems

A parametric convex programming problem with an operator equality constraint and a finite set of functional inequality constraints is considered in a Hilbert space. The instability of this problem and, as a consequence, the instability of the classical Lagrange principle for it is closely related to its regularity and the subdifferentiability properties of the value function in the optimization problem. A sequential Lagrange principle in nondifferential form is proved for the indicated convex programming problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. It is shown that the classical Lagrange principle in this problem can be naturally treated as a limiting variant of its stable sequential counterpart. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimal control problems and inverse problems is discussed. For two illustrative problems of these kinds, the corresponding stable Lagrange principles are formulated in sequential form.     

集值优化问题强有效解的Kuhn Tucker最优性条件

在局部凸空间中考虑集值优化问题（VP）在强有效解意义下的Kuhn-Tucker最优性条件．在近似锥．次类凸假设下利用择一性定理得到了（VP）取得强有效解的必要条件，利用基泛函的性质给出了（VP）取得强有效解的充分条件，最后给出了一种与（VP）等价的无约束规划。     

Attractive force search algorithm for piecewise convex maximization problems

We consider mathematical programming problems with the so-called piecewise convex objective functions. A solution method for this interesting and important class of nonconvex problems is presented. This method is based on Newton??s law of universal gravitation, multicriteria optimization and Helly??s theorem on convex bodies. Numerical experiments using well known classes of test problems on piecewise convex maximization, convex maximization as well as the maximum clique problem show the efficiency of the approach.     

近似锥-次类凸集值优化的严有效性

在Hausdorff局部凸拓扑线性空间中考虑约束集值优化问题(VP)的严有效性.在近似锥-次类凸假设下,利用凸集分离定理,分别得到了Kuhn-Tucker型和Lagrange型最优性条件,建立了与(VP)等价的两种形式的无约束优化.     

Kuhn-Tucker sufficiency for global minimum of multi-extremal mathematical programming problems

The Kuhn-Tucker Sufficiency Theorem states that a feasible point that satisfies the Kuhn-Tucker conditions is a global minimizer for a convex programming problem for which a local minimizer is global. In this paper, we present new Kuhn-Tucker sufficiency conditions for possibly multi-extremal nonconvex mathematical programming problems which may have many local minimizers that are not global. We derive the sufficiency conditions by first constructing weighted sum of square underestimators of the objective function and then by characterizing the global optimality of the underestimators. As a consequence, we derive easily verifiable Kuhn-Tucker sufficient conditions for general quadratic programming problems with equality and inequality constraints. Numerical examples are given to illustrate the significance of our criteria for multi-extremal problems.     

集值优化问题超有效解的Kuhn-Tucker最优性条件

在Hausdorff局部凸拓扑线性空间中考虑约束向量集值优化问题(VP)的超有效性．在近似锥-次类凸假设下,利用择一性定理得到了Kuhn-Tucker型最优性必要条件,利用标量化定理得到了Kuhn-Tucker型最优性充分条件．最后给出了一种与(VP)等价的无约束优化．     

Necessary and sufficient conditions for local and global nondominated solutions in decision problems with multi-objectives

In this paper, the decision problem with multi-objectives is considered, and the nondominated solutions associated with a polyhedral domination cone are discussed. The necessary and sufficient conditions for the solutions are given in the decision space rather than the objective space. The similarity of the solution conditions obtained in this article to the Kuhn-Tucker condition of a convex programming problem is examined. As an application of the solution condition, an algorithm to locate the set of all nondominated solutions is shown for the linear multi-objective decision problem.The author would like to thank the reviewer for his helpful comments.     

Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems

This paper studies a class of multiobjective generalized fractional programming problems, where the numerators of objective functions are the sum of differentiable function and convex function, while the denominators are the difference of differentiable function and convex function. Under the assumption of Calmness Constraint Qualification the Kuhn-Tucker type necessary conditions for efficient solution are given, and the Kuhn-Tucker type sufficient conditions for efficient solution are presented under the assumptions of （F, α, ρ, d）-V-convexity. Subsequently, the optimality conditions for two kinds of duality models are formulated and duality theorems are proved.     

Duality in nonlinear fractional programming

Summary The purpose of the present paper is to introduce, on the lines similar to that ofWolfe [1961], a dual program to a nonlinear fractional program in which the objective function, being the ratio of a convex function to a strictly positive linear function, is a special type of pseudo-convex function and the constraint set is a convex set constrained by convex functions in the form of inequalities. The main results proved are, (i) Weak duality theorem, (ii)Wolfe's (Direct) duality theorem and (iii)Mangasarian's Strict Converse duality theorem.Huard's [1963] andHanson's [1961] converse duality theorems for the present problem have just been stated because they can be obtained as a special case ofMangasarian's theorem [1969, p. 157]. The other important discussion included is to show that the dual program introduced in the present paper can also be obtained throughDinkelbach's Parametric Replacement [1967] of a nonlinear fractional program. Lastly, duality in convex programming is shown to be a special case of the present problem.The present research is partially supported by National Research Council of Canada.     

Regularized optimization methods for convex MINLP problems

We propose regularized cutting-plane methods for solving mixed-integer nonlinear programming problems with nonsmooth convex objective and constraint functions. The given methods iteratively search for trial points in certain localizer sets, constructed by employing linearizations of the involved functions. New trial points can be chosen in several ways; for instance, by minimizing a regularized cutting-plane model if functions are costly. When dealing with hard-to-evaluate functions, the goal is to solve the optimization problem by performing as few function evaluations as possible. Numerical experiments comparing the proposed algorithms with classical methods in this area show the effectiveness of our approach.     

Unboundedness in reverse convex and concave integer programming

In this paper we are concerned with the problem of unboundedness and existence of an optimal solution in reverse convex and concave integer optimization problems. In particular, we present necessary and sufficient conditions for existence of an upper bound for a convex objective function defined over the feasible region contained in ${\mathbb{Z}^n}$ . The conditions for boundedness are provided in a form of an implementable algorithm, showing that for the considered class of functions, the integer programming problem is unbounded if and only if the associated continuous problem is unbounded. We also address the problem of boundedness in the global optimization problem of maximizing a convex function over a set of integers contained in a convex and unbounded region. It is shown in the paper that in both types of integer programming problems, the objective function is either unbounded from above, or it attains its maximum at a feasible integer point.     

Stable iterative Lagrange principle in convex programming as a tool for solving unstable problems

A convex programming problem in a Hilbert space with an operator equality constraint and a finite number of functional inequality constraints is considered. All constraints involve parameters. The close relation of the instability of this problem and, hence, the instability of the classical Lagrange principle for it to its regularity properties and the subdifferentiability of the value function in the problem is discussed. An iterative nondifferential Lagrange principle with a stopping rule is proved for the indicated problem. The principle is stable with respect to errors in the initial data and covers the normal, regular, and abnormal cases of the problem and the case where the classical Lagrange principle does not hold. The possibility of using the stable sequential Lagrange principle for directly solving unstable optimization problems is discussed. The capabilities of this principle are illustrated by numerically solving the classical ill-posed problem of finding the normal solution of a Fredholm integral equation of the first kind.     

Complex Minimax Fractional Programming of Analytic Functions

We prove that a minmax fractional programming problem is equivalent to a minimax nonfractional parametric problem for a given parameter in complex space. Using a parametric approach, we establish the Kuhn-Tucker type necessary optimality conditions and prove the existence theorem of optimality for complex minimax fractional programming in the framework of generalized convexity. Subsequently, we apply the optimality conditions to formulate a one-parameter dual problem and prove weak duality, strong duality, and strict converse duality theorems involving generalized convex complex functions. This research was partly supported by NSC, Taiwan.     

A primal–dual regularized interior-point method for convex quadratic programs

Interior-point methods in augmented form for linear and convex quadratic programming require the solution of a sequence of symmetric indefinite linear systems which are used to derive search directions. Safeguards are typically required in order to handle free variables or rank-deficient Jacobians. We propose a consistent framework and accompanying theoretical justification for regularizing these linear systems. Our approach can be interpreted as a simultaneous proximal-point regularization of the primal and dual problems. The regularization is termedexact to emphasize that, although the problems are regularized, the algorithm recovers a solution of the original problem, for appropriate values of the regularization parameters.