Optimality conditions for piecewiseC 2 nonlinear programming

Several types of finite-dimensional nonlinear programming models are considered in this article. Second-order optimality conditions are derived for these models, under the assumption that the functions involved are piecewiseC ². In rough terms, a real-valued function defined on an open subsetW orR ⁿ is said to be piecewiseC ^k onW if it is continuous onW and if it can be constructed by piecing together onW a finite number of functions of classC ^k .     

On Minimax Fractional Optimality and Duality with Generalized Convexity

In this paper, we establish several sufficient optimality conditions for a class of generalized minimax fractional programming. Based on the sufficient conditions, a new dual model is constructed and duality results are derived. Our study naturally unifies and extends some previously known results in the framework of generalized convexity and dual models. Mathematics Subject Classifications: 90C25, 90C32, 90C47.     

Optimality Conditions and Duality in Nonsmooth Semiinfinite Programming

In this article, we utilize the semiinfinite versions of Guignard's constraint qualification and Motzkin's theorem of the alternative to establish a set of Karush–Kuhn–Tucker-type necessary optimality conditions for a nonsmooth and nonconvex semiinfinite programming problem. Furthermore, we discuss some sufficient optimality conditions and duality relations for our semiinfinite programming problem.     

Optimality conditions in smooth nonlinear programming

This survey is concerned with necessary and sufficient optimality conditions for smooth nonlinear programming problems with inequality and equality constraints. These conditions deal with strict local minimizers of order one and two and with isolated minimizers. In most results, no constraint qualification is required. The optimality conditions are formulated in such a way that the gaps between the necessary and sufficient conditions are small and even vanish completely under mild constraint qualifications.This paper is dedicated to the memory of W. Wetterling.The authors would like to thank Wolfgang Wetterling and Frank Twilt for fruitful discussions and an anonymous referee for many valuable comments.     

Optimality conditions in a step control problem

First-and second-order necessary optimality conditions are obtained for the control of step systems.     

A trust region SQP algorithm for mixed-integer nonlinear programming

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.     

Counterexample and Optimality Conditions in Differentiable Multiobjective Programming

Recently, sufficient optimality theorems for (weak) Pareto-optimal solutions of a multiobjective optimization problem (MOP) were stated in Theorems 3.1 and 3.3 of Ref. 1. In this note, we give a counterexample showing that the theorems of Ref. 1 are not true. Then, by modifying the assumptions of these theorems, we establish two new sufficient optimality theorems for (weak) Pareto-optimal solutions of (MOP); moreover, we give generalized sufficient optimality theorems for (MOP).     

Optimality conditions and duality in subdifferentiable multiobjective fractional programming

Fritz John and Kuhn-Tucker necessary and sufficient conditions for a Pareto optimum of a subdifferentiable multiobjective fractional programming problem are derived without recourse to an equivalent convex program or parametric transformation. A dual problem is introduced and, under convexity assumptions, duality theorems are proved. Furthermore, a Lagrange multiplier theorem is established, a vector-valued ratio-type Lagrangian is introduced, and vector-valued saddle-point results are presented.The authors are thankful to the referees and Professor P. L. Yu for their many useful comments and suggestions which have improved the presentation of the paper.The first author is thankful to the Natural Science and Engineering Research Council of Canada for financial support through Grant No. A-5319. The authors are also thankful to the Dean's Office, Faculty of Management, University of Manitoba, for the financial support provided for the third author's visit to the Faculty.     

Optimality and Duality for a Class of Nondifferentiable Multiobjective Fractional Programming Problems

In this paper, we consider a class of nondifferentiable multiobjective fractional programs in which each component of the objective function contains a term involving the support function of a compact convex set. We establish necessary and sufficient optimality conditions and duality results for weakly efficient solutions of nondifferentiable multiobjective fractional programming problems. This work was supported by Grant R01-2003-000-10825-0 from the Basic Research Program of KOSEF.     

On Global Optimality Conditions and Cutting Plane Algorithms

We discuss global optimality conditions and cutting plane algorithms for DC optimization. The discussion is motivated by certain incorrect results that have appeared recently in the literature on these topics. Incidentally, we investigate the relation of the Tikhonov reciprocity theorem to the optimality conditions for general nonconvex global optimization problems and show how the outer-approximation scheme developed earlier for DC programming can be used to solve a wider class of problems.     

Errata Corrige: On an Optimality Condition in DC Optimization

We take care of an omission in Proposition 4.4 of Ref. 1.     

Necessary Optimality Conditions for Problems with Equality and Inequality Constraints: Abnormal Case

This article concerns second-order necessary conditions for an abnormal local minimizer of a nonlinear optimization problem with equality and inequality constraints. The obtained optimality conditions improve the ones available in the literature in that the associated set of Lagrange multipliers is the smallest possible. The first and the second authors were supported by Russian Foundation of Basic Research, Projects 08-01-90267, 08-01-90001. The second and third authors were supported by FCT (Portugal), Research Projects SFRH/BPD/26231/2006, PTDC/EEA-ACR/75242/2006.     

Interior-point methods for nonconvex nonlinear programming: regularization and warmstarts

In this paper, we investigate the use of an exact primal-dual penalty approach within the framework of an interior-point method for nonconvex nonlinear programming. This approach provides regularization and relaxation, which can aid in solving ill-behaved problems and in warmstarting the algorithm. We present details of our implementation within the loqo algorithm and provide extensive numerical results on the CUTEr test set and on warmstarting in the context of quadratic, nonlinear, mixed integer nonlinear, and goal programming. Research of the first author is sponsored by ONR grant N00014-04-1-0145. Research of the second author is supported by NSF grant DMS-0107450.     

Optimality conditions and newton-type methods for mathematical programs with vanishing constraints

A new class of optimization problems is discussed in which some constraints must hold in certain regions of the corresponding space rather than everywhere. In particular, the optimal design of topologies for mechanical structures can be reduced to problems of this kind. Problems in this class are difficult to analyze and solve numerically because their constraints are usually irregular. Some known first- and second-order necessary conditions for local optimality are refined for problems with vanishing constraints, and special Newton-type methods are developed for solving such problems.     

Constraint incorporation in optimization

Numerical methods for solving constrained optimization problems need to incorporate the constraints in a manner that satisfies essentially competing interests; the incorporation needs to be simple enough that the solution method is tractable, yet complex enough to ensure the validity of the ultimate solution. We introduce a framework for constraint incorporation that identifies a minimal acceptable level of complexity and defines two basic types of constraint incorporation which (with combinations) cover nearly all popular numerical methods for constrained optimization, including trust region methods, penalty methods, barrier methods, penalty-multiplier methods, and sequential quadratic programming methods. The broad application of our framework relies on addition and chain rules for constraint incorporation which we develop here.     

随机系数下线性二次控制问题的最优性条件及其应用*

本文旨在研究随机系数下随机微分方程的线性二次最优控制问题.本文从闭环最优控制/策略存在的必要性条件的角度开展研究. 若闭环最优控制/策略存在, 得到其显示反馈表示、带伪逆运算的倒向随机Riccati方程的适定性及不同系数间满足的一些本质性条件. 此处结论本质地推广和改进了文[Ait Rami M, Moore J, Zhou X. Indefinite stochastic linear quadratic control and generalized differential Riccati equation [J]. {\it SIAM J Control Optim,} 2001, 40:1296--1311;Sun J, Yong J. Linear quadratic stochastic differential games: open-loop and closed-loop saddle points [J]. {\it SIAM J Control Optim,} 2014, 52:4082--4121;L\"{u} Q, Wang T, Zhang X. Characterization of optimal feedback for stochastic linear quadratic control problems,Probab Uncertain Quant Risk, 2017, 2017, 2:11, DOI 10.1186/s41546-017-0022-7]的相应结论.此外, 本文得到了一个关于倒向随机Riccati方程和二阶伴随方程两类方程适应解之间的微妙关系. 注意到,这一结论在现有文献中首次出现. 最后, 本文讨论了在均值方差对冲问题中的应用.     

Optimality conditions for nonconvex semidefinite programming

This paper concerns nonlinear semidefinite programming problems for which no convexity assumptions can be made. We derive first- and second-order optimality conditions analogous to those for nonlinear programming. Using techniques similar to those used in nonlinear programming, we extend existing theory to cover situations where the constraint matrix is structurally sparse. The discussion covers the case when strict complementarity does not hold. The regularity conditions used are consistent with those of nonlinear programming in the sense that the conventional optimality conditions for nonlinear programming are obtained when the constraint matrix is diagonal. Received: May 15, 1998 / Accepted: April 12, 2000?Published online May 12, 2000     

Refinements of necessary conditions for optimality in nonlinear programming

In this paper, a new set of necessary conditions for optimality is introduced with reference to the differentiable nonlinear programming problem. It is shown that these necessary conditions are sharper than the usual Fritz John ones. A constraint qualification relevant to the new necessary conditions is defined and extensions to the locally Lipschitz case are presented.     

Optimality Conditions and Duality in Nondifferentiable Minimax Fractional Programming with Generalized Convexity

We establish sufficient optimality conditions for a class of nondifferentiable minimax fractional programming problems involving (F, α, ρ, d)-convexity. Subsequently, we apply the optimality conditions to formulate two types of dual problems and prove appropriate duality theorems. The authors thank the referee for valuable suggestions improving the presentation of the paper.     

First and second order analysis of nonlinear semidefinite programs

In this paper we study nonlinear semidefinite programming problems. Convexity, duality and first-order optimality conditions for such problems are presented. A second-order analysis is also given. Second-order necessary and sufficient optimality conditions are derived. Finally, sensitivity analysis of such programs is discussed.