Optimality and Duality for Complex Nondifferentiable Fractional Programming

Both parametric and nonparametric necessary and sufficient optimality conditions are established for a class of complex nondifferentiable fractional programming problems containing generalized convex functions. Subsequently, these optimality criteria are utilized as a basis for constructing one parametric and two other parameter-free dual models with appropriate duality theorems.     

多目标分式变分问题的混合对偶性

在广义B-Ⅰ凸性条件下,建立了多目标分式变分问题的混合对偶模型,使得M ond-W e ir型对偶和W o lfe型成为其特殊情况,并建立了关于有效解的混合对偶理论.     

On duality in linear fractional programming

In this paper, a dual of a given linear fractional program is defined and the weak, direct and converse duality theorems are proved. Both the primal and the dual are linear fractional programs. This duality theory leads to necessary and sufficient conditions for the optimality of a given feasible solution. A unmerical example is presented to illustrate the theory in this connection. The equivalence of Charnes and Cooper dual and Dinkelbach’s parametric dual of a linear fractional program is also established.     

Nonlinear Mathieu equation and coupled resonance mechanism

The method of multiple-scales is used to determine a third-order solution for a cubic nonlinear Mathieu equation. The perturbation solutions are imposed on the so-called solvability conditions. Solvability conditions in the non-resonance case yield the standard Landau equation. Several types of a parametric Landau equation are derived in the neighborhood of five different resonance cases. These parametric Landau equations contain a parametric complex conjugate term or a parametric second-order complex conjugate term or a parametric complex conjugate term as well as a parametric second-order term. Necessary and sufficient conditions for stability are performed in each resonance case. Stability criteria correspond to each parametric Landau equation and are derived by linear perturbation. Stability criteria for the non-trivial steady-state response are discussed. The analysis leads to simultaneous resonance. Transition curves are performed in each case. Numerical calculations are made for some transition curves to illustrate the coupled resonance regions, where the induced stability tongues within the instability tongues are observed. The amplitude of the periodic coefficient of Mathieu equation plays a dual role in the stability criteria for nonlinear Mathieu equation.     

Duality of Nonscalarized Multiobjective Linear Programs: Dual Balance,Level Sets,and Dual Clusters of Optimal Vectors

A new concept of duality is proposed for multiobjective linear programs. It is based on a set expansion process for the computation of optimal solutions without scalarization. The duality gap qualifications are investigated; the primal–dual balance set and level set equations are derived. It is demonstrated that the nonscalarized dual problem presents a cluster of optimal dual vectors that corresponds to a unique optimal primal vector. Comparisons are made with linear utility, minmax and minmin scalarizations. Connections to Pareto optimality are studied and relations to sensitivity and parametric programming are discussed. The ideas are illustrated by examples.     

Optimality and duality in nonsmooth multiobjective fractional programming problem with constraints

In this paper, we establish some quotient calculus rules in terms of contingent derivatives for the two extended-real-valued functions defined on a Banach space and study a nonsmooth multiobjective fractional programming problem with set, generalized inequality and equality constraints. We define a new parametric problem associated with these problem and introduce some concepts for the (local) weak minimizers to such problems. Some primal and dual necessary optimality conditions in terms of contingent derivatives for the local weak minimizers are provided. Under suitable assumptions, sufficient optimality conditions for the local weak minimizers which are very close to necessary optimality conditions are obtained. An application of the result for establishing three parametric, Mond–Weir and Wolfe dual problems and several various duality theorems for the same is presented. Some examples are also given for our findings.

     

A Branch-and-Bound Approach for Solving a Class of Generalized Semi-infinite Programming Problems

A nonconvex generalized semi-infinite programming problem is considered, involving parametric max-functions in both the objective and the constraints. For a fixed vector of parameters, the values of these parametric max-functions are given as optimal values of convex quadratic programming problems. Assuming that for each parameter the parametric quadratic problems satisfy the strong duality relation, conditions are described ensuring the uniform boundedness of the optimal sets of the dual problems w.r.t. the parameter. Finally a branch-and-bound approach is suggested transforming the problem of finding an approximate global minimum of the original nonconvex optimization problem into the solution of a finite number of convex problems.     

High-order dual-parametric finite element methods for cavitation computation in nonlinear elasticity

In this paper, we present the numerical analysis on high order dual parametric finite element methods for the cavitation computation problems in nonlinear elasticity, which leads to a meshing strategy assuring high efficiency on numerical approximations to cavity deformations. Furthermore, to cope with the high order approximation of the finite element methods, properly chosen weighted Gaussian type numerical quadrature is applied to the singular part of the elastic energy. Our numerical experiments show that the high order dual parametric finite element methods work well when coupled with properly designed weighted Gaussian type numerical quadratures for the singular part of the elastic energy, and the convergence rates of the numerical cavity solutions are shown to be significantly improved as expected.     

Using the parametric approach to solve the continuous-time linear fractional max–min problems

A numerical algorithm based on parametric approach is proposed in this paper to solve a class of continuous-time linear fractional max-min programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as a parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this algorithm.     

Minimax linear programming problem

In this paper, we consider the following minimax linear programming problem: min z = max_{1 ≤ j ≤ n}{C_jX_j}, subject to Ax = g, x ≥ 0. It is well known that this problem can be transformed into a linear program by introducing n additional constraints. We note that these additional constraints can be considered implicitly by treating them as parametric upper bounds. Based on this approach we develop two algorithms: a parametric algorithm and a primal—dual algorithm. The parametric algorithm solves a linear programming problem with parametric upper bounds and the primal—dual algorithm solves a sequence of related dual feasible linear programming problems. Computation results are also presented, which indicate that both the algorithms are substantially faster than the simplex algorithm applied to the enlarged linear programming problem.     

Generalized sensitivity analysis of nonlinear programs using a sequence of quadratic programs

ABSTRACT

Local sensitivity information is obtained for KKT points of parametric NLPs that may exhibit active set changes under parametric perturbations; under appropriate regularity conditions, computationally relevant generalized derivatives of primal and dual variable solutions of parametric NLPs are calculated. Ralph and Dempe obtained directional derivatives of solutions of parametric NLPs exhibiting active set changes from the unique solution of an auxiliary quadratic program. This article uses lexicographic directional derivatives, a newly developed tool in nonsmooth analysis, to generalize the classical NLP sensitivity analysis theory of Ralph and Dempe. By viewing said auxiliary quadratic program as a parametric NLP, the results of Ralph and Dempe are applied to furnish a sequence of coupled QPs, whose unique solutions yield generalized derivative information for the NLP. A practically implementable algorithm is provided. The theory developed here is motivated by widespread applications of nonlinear programming sensitivity analysis, such as in dynamic control and optimization problems.     

Dimension invariance of finite frames of translates and Gabor frames

A dimension invariance property for finite frames of translates and Gabor frames is discussed. Under appropriate support conditions among the frame and dual frame generating functions, we show that a pair of dual frames evaluated in a given space remains a valid dual set if they are naturally embedded in the underlying space of almost arbitrarily enlarged dimension. Consequently, the evaluation of duals in a very large dimensional space is now easily accessible by merely working in a space of some much smaller dimension. A number of uniform and non-uniform schemes are studied. To satisfy the support conditions, a method of finding valid alternate dual functions with small support via a known parametric dual frame formula is discussed. Oftentimes it is convenient to have truncated approximate duals that satisfy the support conditions. Stability studies of the dimension invariance principle via such approximate duals are also presented.     

Using the Dinkelbach-type algorithm to solve the continuous-time linear fractional programming problems

A Dinkelbach-type algorithm is proposed in this paper to solve a class of continuous-time linear fractional programming problems. We shall transform this original problem into a continuous-time non-fractional programming problem, which unfortunately happens to be a continuous-time nonlinear programming problem. In order to tackle this nonlinear problem, we propose the auxiliary problem that will be formulated as parametric continuous-time linear programming problem. We also introduce a dual problem of this parametric continuous-time linear programming problem in which the weak duality theorem also holds true. We introduce the discrete approximation method to solve the primal and dual pair of parametric continuous-time linear programming problems by using the recurrence method. Finally, we provide two numerical examples to demonstrate the usefulness of this practical algorithm.     

A dual view of equilibrium problems

The aim of this paper is to introduce and study a dual problem associated to a generalized equilibrium problem (GEP). We show that the solutions of (GEP) and its dual are strictly related to the saddle points of an associated Lagrangian function, and, under some suitable conditions, to the solutions of a family of parametric optimization problems and their dual problems. Our results allow us to show that well-known concepts and results from duality theory of some important particular cases of (GEP) like variational inequalities and optimization problems can be recovered.     

Parametric mortality improvement rate modelling and projecting

We investigate the modelling of mortality improvement rates and the feasibility of projecting mortality improvement rates (as opposed to projecting mortality rates), using parametric predictor structures that are amenable to simple time series forecasting. This leads to our proposing a parallel dual approach to the direct parametric modelling and projecting of mortality rates. Comparisons of simulated life expectancy predictions (by the cohort method) using the England and Wales population mortality experiences for males and females under a variety of controlled data trimming exercises are presented in detail and comparisons are also made between the parallel modelling approaches.     

Convex Difference Criteria for the Quantitative Stability of Parametric Quasidifferentiable Systems

In this paper solvability and Lipschitzian stability properties for a special class of nonsmooth parametric generalized systems defined in Banach are studied via a variational analysis approach. Verifiable sufficient conditions for such properties to hold under scalar quasidifferentiability assumptions are formulated by combining *-difference and Demyanov difference of convex compact subsets of the dual space with classic quasidifferential calculus constructions. Applications to the formulation of sufficient conditions for metric regularity/open covering of nonsmooth maps, along with their employment in deriving optimality conditions for quasidifferentiable extremum problems, as well as an application to the study of semicontinuity of the optimal value function in parametric optimization are discussed. In memory of Aleksandr Moiseevich Rubinov (1940–2006).     

A general parametric analysis approach and its implication to sensitivity analysis in interior point methods

Adler and Monteiro (1992) developed a parametric analysis approach that is naturally related to the geometry of the linear program. This approach is based on the availability of primal and dual optimal solutions satisfying strong complementarity. In this paper, we develop an alternative geometric approach for parametric analysis which does not require the strong complementarity condition. This parametric analysis approach is used to develop range and marginal analysis techniques which are suitable for interior point methods. Two approaches are developed, namely the LU factorization approach and the affine scaling approach. Presented at the ORSA/TIMS, Nashville, TN, USA, May 1991. Supported by the National Science Foundation (NSF) under Grant No. DDM-9109404 and Grant No. DMI-9496178. This work was done while the author was a faculty member of the Systems and Industrial Engineering Department at The University of Arizona. Supported in part by the GTE Laboratories and the National Science Foundation (NSF) under Grant No. CCR-9019469.     

Parametric Disjunctive Programming: One-Sided Differentiability of the Value Function

We consider a countable family of one-parameter convex programs and give sufficient conditions for the one-sided differentiability of its optimal value function. The analysis is based on the Borwein dual problem for a family of convex programs (a convex disjunctive program). We give conditions that assure stability of the situation of perfect duality in the Borwein theory.For the reader's convenience, we start with a review of duality results for families of convex programs. A parametric family of dual problems is introduced that contains the dual problems of Balas and Borwein as special cases. In addition, a vector optimization problem is defined as a dual problem. This generalizes a result by Helbig about families of linear programs.     

OPTIMALITY CONDITIONS AND DUALITY MODELS FOR A CLASS OF NONSMOOTH CONTINUOUS-TIME GENERALIZED FRACTIONAL PROGRAMMING PROBLEMS

Abstract

Both parametric and parameter-free stationary-point-type and saddle-point-type necessary and sufficient optimality conditions are established for a class of nonsmooth continuous-time generalized fractional programming problems with Volterra-type integral inequality and nonnegativity constraints. These optimality criteria are then utilized for constructing ten parametric and parameter-free Wolfe-type and Lagrangian-type dual problems and for proving weak, strong, and strict converse duality theorems. Furthermore, it is briefly pointed out how similar optimality and duality results can be obtained for two important special cases of the main problem containing arbitrary norms and square roots of positive semidefinite quadratic forms. All the results developed here are also applicable to continuous-time programming problems with fractional, discrete max, and conventional objective functions, which are special cases of the main problem studied in this paper.