Partial linearization methods in nonlinear programming

In this paper, we characterize a class of feasible direction methods in nonlinear programming through the concept of partial linearization of the objective function. Based on a feasible point, the objective is replaced by an arbitrary convex and continuously differentiable function, and the error is taken into account by a first-order approximation of it. A new feasible point is defined through a line search with respect to the original objective, toward the solution of the approximate problem. Global convergence results are given for exact and approximate line searches, and possible interpretations are made. We present some instances of the general algorithm and discuss extensions to nondifferentiable programming.The author wishes to thank Drs. K. Holmberg, T. Larsson, and A. Migdalas for their helpful comments.     

求广义几何规划全局最优解的线性化方法

对广义几何规划问题(GGP)提出了一个确定型全局优化算法，这类优化问题能广泛应用于工程设计和非线性系统的鲁棒稳定性分析等实际问题中，使用指数变换及对目标函数和约束函数的线性下界估计，建立了GGP的松弛线性规划(RLP)，通过对RLP可行域的细分以及一系列RLP的求解过程，从理论上证明了算法能收敛到GGP的全局最优解，对一个化学工程设计问题应用本文算法，数值实验表明本文方法是可行的。     

Constrained ratio games and generalized fractional programming

A certain constrained ratio game is shown to be equivalent to a pair of mutually dual generalized fractional programming problems. This also extends the concept of symmetric duality to min-max fractional programming.Research by this author was carried out while he was visiting I.I.T. Delhi, India, under an Australian Vice-Chancellors' Committee Visiting Fellowship.     

Multiobjective fractional programming with generalized convexity

This paper derives several results regarding the optimality conditions and duality properties for the class of multiobjective fractional programs under generalized convexity assumptions. These results are obtained by applying a parametric approach to reduce the problem to a more conventional form.     

Convex programming with set-inclusive constraints and its applications to generalized linear and fractional programming

Duality results are established in convex programming with the set-inclusive constraints studied by Soyster. The recently developed duality theory for generalized linear programs by Thuente is further generalized and also brought into the framework of Soyster's theory. Convex programming with set-inclusive constraints is further extended to fractional programming.     

一类多目标广义分式规划问题的最优性条件和对偶

研究了一类不可微多目标广义分式规划问题．首先,在广义Abadie约束品性条件下,给出了其真有效解的Kuhn—Tucker型必要条件．随后,在（C,a,P,d）一凸性假设下给出其真有效解的充分条件．最后,在此基础上建立了一种对偶模型,证明了对偶定理．得到的结果改进了相关文献中的相应结论．     

一种新广义凸多目标分式规划的最优性充分条件

提出了（F,α,ρ,θ）-b-凸函数的概念,它是一类新的广义凸函数,并给出了这类广义凸函数的性质.在此基础上,讨论了目标函数和约束函数均为（F,α,ρ,θ）-b-凸函数的多目标分式规划,利用广义K-T条件,得到了这类多目标规划有效解和弱有效解的几个充分条件,推广了已有文献的相关结果.     

Proximal-type methods with generalized Bregman functions and applications to generalized fractional programming

We analyse proximal-type minimization methods with generalized Bregman functions by considering a general scheme based on the one studied by Kiwiel [K.C. Kiwiel, Proximal minimization methods with generalized Bregman functions, SIAM J. Control Optim. 35(4) (1997), pp. 1142–1168.] and on successive approximation methods. We apply this scheme to construct methods for generalized fractional programmes.     

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.     

Conjugate duality in generalized fractional programming

The concepts of conjugate duality are used to establish dual programs for a class of generalized nonlinear fractional programs. It is now known that, under certain restrictions, a symmetric duality exists for generalized linear fractional programs. In this paper, we establish this symmetric duality for the nonlinear case.     

Using duality to solve generalized fractional programming problems

In this paper we explore the relations between the standard dual problem of a convex generalized fractional programming problem and the partial dual problem proposed by Barros et al. (1994). Taking into account the similarities between these dual problems and using basic duality results we propose a new algorithm to directly solve the standard dual of a convex generalized fractional programming problem, and hence the original primal problem, if strong duality holds. This new algorithm works in a similar way as the algorithm proposed in Barros et al. (1994) to solve the partial dual problem. Although the convergence rates of both algorithms are similar, the new algorithm requires slightly more restrictive assumptions to ensure a superlinear convergence rate. An important characteristic of the new algorithm is that it extends to the nonlinear case the Dinkelbach-type algorithm of Crouzeix et al. (1985) applied to the standard dual problem of a generalized linear fractional program. Moreover, the general duality results derived for the nonlinear case, yield an alternative way to derive the standard dual of a generalized linear fractional program. The numerical results, in case of quadratic-linear ratios and linear constraints, show that solving the standard dual via the new algorithm is in most cases more efficient than applying directly the Dinkelbach-type algorithm to the original generalized fractional programming problem. However, the numerical results also indicate that solving the alternative dual (Barros et al., 1994) is in general more efficient than solving the standard dual.This research was carried out at the Econometric Institute, Erasmus University Rotterdam, the Netherlands and was supported by the Tinbergen Institute Rotterdam     

Saddle-point type optimality criteria for generalized fractional programming

Generally speaking, one cannot expect that there exists a Lagrangian saddle point even for a linear fractional program. Using the functionsGF(x,r ₀,r) andGK(x,u) somewhat like the Lagrangian functions, we present saddle-point type optimality criteria for generalized fractional programming under carefully selected assumptions. In addition, we point out conditions that ensure that a local optimal solution of the program mentioned is global.The author wishes to thank an anonymous referee for his detailed comments toward a clear presentation of this paper.     

Generalized linear multiplicative and fractional programming

This paper addresses itself to the algorithm for minimizing the sum of a convex function and a product of two linear functions over a polytope. It is shown that this nonconvex minimization problem can be solved by solving a sequence of convex programming problems. The basic idea of this algorithm is to embed the original problem into a problem in higher dimension and apply a parametric programming (path following) approach. Also it is shown that the same idea can be applied to a generalized linear fractional programming problem whose objective function is the sum of a convex function and a linear fractional function.     

A duality theorem for a homogeneous fractional programming problem

The usual theory of duality for linear fractional programs is extended by replacing the linear functions in the numerator and denominator by arbitrary positively homogeneous convex functions. In the constraints, the positive orthant inR ⁿ is replaced by an arbitrary cone. The resultant duality theorem contains a recent result of Chandra and Gulati as a special case.The authors wish to thank the referee for a number of valuable suggestions, particularly improvements in Theorem 3.4 and Corollary 3.1.     

Global optimization for generalized geometric programming problems with discrete variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management, but most existing methods for solving GGP actually only consider continuous variables. This article presents a new branch-and-bound algorithm for globally solving GGP problems with discrete variables. For minimizing the problem, an equivalent monotonic optimization problem (P) with discrete variables is presented by exploiting the special structure of GGP. In the algorithm, the lower bounds are computed by solving ordinary linear programming problems that are derived via a linearization technique. In contrast to pure branch-and-bound methods, the algorithm can perform a domain reduction cut per iteration by using the monotonicity of problem (P), which can suppress the rapid growth of branching tree in the branch-and-bound search so that the performance of the algorithm is further improved. Computational results for several sample examples and small randomly generated problems are reported to vindicate our conclusions.     

Duality and saddle-point type optimality for generalized nonlinear fractional programming

In this paper, we establish two theorems of alternative with generalized subconvexlikeness. We introduce two dual models for a generalized fractional programming problem. Theorems of alternative are then applied to establish duality theorems and a saddle-point type optimality condition.     

An improved linearization strategy for zero-one quadratic programming problems

We present a new linearized model for the zero-one quadratic programming problem, whose size is linear in terms of the number of variables in the original nonlinear problem. Our derivation yields three alternative reformulations, each varying in model size and tightness. We show that our models are at least as tight as the one recently proposed in [7], and examine the theoretical relationship of our models to a standard linearization of the zero-one quadratic programming problem. Finally, we demonstrate the efficacy of solving each of these models on a set of randomly generated test instances.     

Strong duality for robust minimax fractional programming problems

We develop a duality theory for minimax fractional programming problems in the face of data uncertainty both in the objective and constraints. Following the framework of robust optimization, we establish strong duality between the robust counterpart of an uncertain minimax convex–concave fractional program, termed as robust minimax fractional program, and the optimistic counterpart of its uncertain conventional dual program, called optimistic dual. In the case of a robust minimax linear fractional program with scenario uncertainty in the numerator of the objective function, we show that the optimistic dual is a simple linear program when the constraint uncertainty is expressed as bounded intervals. We also show that the dual can be reformulated as a second-order cone programming problem when the constraint uncertainty is given by ellipsoids. In these cases, the optimistic dual problems are computationally tractable and their solutions can be validated in polynomial time. We further show that, for robust minimax linear fractional programs with interval uncertainty, the conventional dual of its robust counterpart and the optimistic dual are equivalent.     

Ranking in integer linear fractional programming problems

An algorithm is developed which ranks the feasible solutions of an integer fractional programming problem in decreasing order of the objective function values.

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Zusammenfassung Es wird ein Algorithmus angegeben, der die zulässigen Lösungen eines ganzzahligen Quotientenprogrammes nach fallenden Zielfunktionswerten liefert.

     

Bicriteria linear fractional programming

As a step toward the investigation of the multicriteria linear fractional program, this paper provides a thorough analysis of the bicriteria case. It is shown that the set of efficient points is a finite union of linearly constrained sets and the efficient frontier is the image of a finite number of connected line segments of efficient points. A simple algorithm using only one-dimensional parametric linear programming techniques is developed to evaluate the efficient frontier.This research was partially supported by NRC Research Grant No. A4743. The authors wish to thank two anonymous referees for their helpful comments on an earlier draft of this paper.