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.     

Farkas-type results for fractional programming problems

Considering a constrained fractional programming problem, within the present paper we present some necessary and sufficient conditions, which ensure that the optimal objective value of the considered problem is greater than or equal to a given real constant. The desired results are obtained using the Fenchel–Lagrange duality approach applied to an optimization problem with convex or difference of convex (DC) objective functions and finitely many convex constraints. These are obtained from the initial fractional programming problem using an idea due to Dinkelbach. We also show that our general results encompass as special cases some recently obtained Farkas-type results.     

Duality for semi-definite and semi-infinite programming

In this article, we study semi-definite and semi-infinite programming problems (SDSIP), which includes semi-infinite linear programs and semi-definite programs as special cases. We establish that a uniform duality between the homogeneous (SDSIP) and its Lagrangian-type dual problem is equivalent to the closedness condition of certain cone. Moreover, this closedness condition was assured by a generalized canonically closedness condition and a Slater condition. Corresponding results for the nonhomogeneous (SDSIP) problem were obtained by transforming it into an equivalent homogeneous (SDSIP) problem.     

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.     

Optimality and Duality for Invex Nonsmooth Multiobjective programming problems

We consider nonsmooth multiobjective programming problems with inequality and equality constraints involving locally Lipschitz functions. Several sufficient optimality conditions under various (generalized) invexity assumptions and certain regularity conditions are presented. In addition, we introduce a Wolfe-type dual and Mond–Weir-type dual and establish duality relations under various (generalized) invexity and regularity conditions.     

A sequential quadratic penalty method for nonlinear semidefinite programming

In this article, a nonlinear semidefinite program is reformulated into a mathematical program with a matrix equality constraint and a sequential quadratic penalty method is proposed to solve the latter problem. We discuss the differentiability and convexity of the penalty function. Necessary and sufficient conditions for the convergence of optimal values of penalty problems to that of the original semidefinite program are obtained. The convergence of optimal solutions of penalty problems to that of the original semidefinite program is also investigated. We show that any limit point of a sequence of stationary points of penalty problems satisfies the KKT optimality condition of the semidefinite program. Smoothed penalty problems that have the same order of smothness as the original semidefinite program are adopted. Corresponding results such as the convexity of the smoothed penalty function, the convergence of optimal values, optimal solutions and the stationary points of the smoothed penalty problems are obtained.     

A Note on the Borsuk Non-Retraction Theorem

We give an easy proof of a result in [1] generalizing the well-known Borsuks non-retraction theorem.     

A general class of penalty/barrier path-following Newton methods for nonlinear programming

In the present article rather general penalty/barrier-methods (e.g. logarithmic barriers, SUMT, exponential penalties), which define a local continuously differentiable primal and dual path, are analyzed in case of strict local minima of nonlinear problems with inequality as well as equality constraints. In particular, the radius of convergence of Newton's method depending on the penalty/barrier-parameter is estimated. Unlike using self-concordance properties, the convergence bounds are derived by direct estimations of the solutions of the Newton equations. By means of the obtained results parameter selection rules are studied which guarantee the local convergence of the considered penalty/barrier-techniques with only a finite number of Newton steps at each parameter level. Numerical examples illustrate the practical behavior of the proposed class of methods.     

Optimality conditions and duality for a class of nondifferentiable multi-objective fractional programming problems

A class of multi-objective fractional programming problems (MFP) are considered where the involved functions are locally Lipschitz. In order to deduce our main results, we give the definition of the generalized (F,θ,ρ,d)-convex class about the Clarke’s generalized gradient. Under the above generalized convexity assumption, necessary and sufficient conditions for optimality are given. Finally, a dual problem corresponding to (MFP) is formulated, appropriate dual theorems are proved.      

Optimality conditions and duality for a minimax fractional programming with generalized convexity

We establish the sufficient conditions for generalized fractional programming from a viewpoint of the generalized convexity. When the sufficient conditions are utilized, the corresponding duality theorems are derived for two types of duals of the generalized fractional programming. We extend the corresponding results of several authors.     

Analysis of a Quantum Subband Model for the Transport of Partially Confined Charged Particles

This paper is devoted to the analysis of a quantum subband model, which is presented as an alternative to the standard 3D Schr?dinger-Poisson system for modeling the transport of electrons strongly confined along one direction. This subband model is composed of quasistatic 1D Schr?dinger equations in the direction of the confinement, coupled to 2D time-dependent Schr?dinger equations describing the transport in the non-confined directions. Selfconsistent electrostatic interactions are also taken into account via the Poisson equation. This system is studied in the framework of the strong partial confinement asymptotics introduced in the article “Adiabatic approximation of the Schr?dinger-Poisson system with a partial confinement”, by Ben Abdallah, Méhats and Pinaud (SIAM J. Math. Anal. 36 (2005), 986–1013).     

Two-Orbit Kähler Manifolds and Morse Theory

We deal with compact Kähler manifolds M acted on by a compact Lie group K of isometries, whose complexification K has exactly one open and one closed orbit in M. If the K-action is Hamiltonian, we investigate topological and cohomological properties of M.     

An analysis of some dual problems in multiobjective optimization (I)

In this work we study the duality for a general multiobjective optimization problem. Considering, first, a scalar problem, different duals using the conjugacy approach are presented. Starting from these scalar duals, we introduce six different multiobjective dual problems to the primal one, one depending on certain vector parameters. The existence of weak and, under certain conditions, strong duality between the primal and the dual problems is shown. Afterwards, some inclusion results for the image sets of the multiobjective dual problems (D ₁), (D _α) and (D_FL ) are derived. Moreover, we verify that the efficiency sets within the image sets of these problems coincide, but the image sets themselves do not.     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

A generalized karush-kuhn-tucki optimality condition without constraint qualification using tl approximate subdifferential

A generalized Karush-Kuhn-Tucker first order optimality condition is established for an abstract cone-constrained programming problem involving locally Lipschitz functions using the approximate subdifferential. This result is obtained without recourse to a constraint qualification by imposing additional generalized convexity conditions on the constraint functions. A new Fritz John optimality condition is developed as a precursor to the main result. Several examples are provided to illustrate the results along with a discussion of applications to concave minimization problems and to stochastic programming problems with nonsmooth data.     

Multistage bilevel programming problems

In this article, we first examine some modeling scenarios for a multistage bilevel programming problem and develop the solution techniques based on certain reformulations of the original problem. The optimality conditions obtained for a class of multistage problems are given in terms of the second order subdifferentials of Mordukhovich.     

Wavelet Frames on Groups and Hypergroups via Discretization of Calderón Formulas

Continuous wavelets are often studied in the general framework of representation theory of square-integrable representations, or by using convolution relations and Fourier transforms. We consider the well-known problem whether these continuous wavelets can be discretized to yield wavelet frames. In this paper we use Calderón-Zygmund singular integral operators and atomic decompositions on spaces of homogeneous type, endowed with families of general translations and dilations, to attack this problem, and obtain strong convergence results for wavelets expansions in a variety of classical functional spaces and smooth molecule spaces. This approach is powerful enough to yield, in a uniform way, for example, frames of smooth wavelets for matrix dilations in ⁿ, for an affine extension of the Heisenberg group, and on many commutative hypergroups.     

A New Regularity Criterion for the Navier-Stokes Equations in Terms of the Direction of Vorticity

In this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations.     

Retraction of a Compact Subspace onto Its Boundary

 Generalizing the famous Borsuk’s non-retraction theorem, it is proved that there is a retraction from a compact subspace of a euclidean space onto its boundary if and only if it has an empty interior. Received October 31, 2001; in revised form March 14, 2002 Published online September 9, 2002