Duality Results Research Articles

Joint Information and Jamming Beamforming for Secrecy Rate Maximization in Cognitive Radio Networks

In this paper, we consider the secure beamforming design for an underlay cognitive radio multiple-input single-output broadcast channel in the presence of multiple passive eavesdroppers. Our goal is to design a jamming noise (JN) transmit strategy to maximize the secrecy rate of the secondary system. By utilizing the zero-forcing method to eliminate the interference caused by JN to the secondary user, we study the joint optimization of the information and JN beamforming for secrecy rate maximization of the secondary system while satisfying all the interference power constraints at the primary users, as well as the per-antenna power constraint at the secondary transmitter. For an optimal beamforming design, the original problem is a nonconvex program, which can be reformulated as a convex program by applying the rank relaxation method. To this end, we prove that the rank relaxation is tight and propose a barrier interior-point method to solve the resulting saddle point problem based on a duality result. To find the global optimal solution, we transform the considered problem into an unconstrained optimization problem. We then employ Broyden–Fletcher–Goldfarb–Shanno method to solve the resulting unconstrained problem, which helps reduce the complexity significantly, compared with the conventional methods. Simulation results show the fast convergence of the proposed algorithm and substantial performance improvements over the existing approaches.

An Efficient Zero-Forcing Precoding Design for Cognitive MIMO Broadcast Channels

We consider linear precoding design for an underlay cognitive radio multiple-input multiple-output broadcast channel in the presence of multiple primary users (PUs). Under the assumption of imperfect channel state information (CSI) of the PUs, the objective of this letter is to maximize the sum rate of the secondary system, subject to the power budget at the secondary base station and the interference power constraints at the PUs. The design problem is non-convex, and thus is difficult to solve in general. Herein, we first convert a non-convex constraint related to the imperfect CSI of the PUs into a convex constraint, and then invoke a rank relaxation method to transform the considered problem into a convex–concave problem based on a downlink–uplink duality result. Simulation results are provided to demonstrate the effectiveness and robustness of the proposed design against CSI imperfection.

OPTIMALITY AND PARAMETRIC DUALITY FOR NONSMOOTH MINIMAX FRACTIONAL PROGRAMMING PROBLEMS INVOLVING L-INVEX-INFINE FUNCTIONS

The Karush-Kuhn-Tucker type necessary optimality conditions are given for the nonsmooth minimax fractional programming problem with inequality and equality constraints. Subsequently, based on the idea of L-invex-infine functions defined in terms of the limiting/Mordukhovich subdifferential of locally Lipschitz functions, we obtain sufficient optimality conditions for the considered nonsmooth minimax fractional programming problem and also we provide an example to justify the existence of sufficient optimality conditions. Furthermore, we propose a parametric type dual problem and explore duality results.

Eisenberg's Duality in Homogeneous Programming, Shephard's Duality and Economic Analysis

AbstractThis note is to reintroduce to the reader Eisenberg's symmetric duality theorem in homogeneous programming problems as a useful tool in economic analysis, and thereby to pay a due tribute to him for one of his mathematical contributions. His duality result has been almost in oblivion during the development of Shephard's duality theory between cost and production, and has seldom been mentioned in the literature about the dualities concerning Shephard's distance function, Luenberger's benefit function and directional distance functions proposed by many authors. We show that from Eisenberg's duality it is possible to derive in a systematic way these dualities so far obtained. We also present a further extension of the duality for generalized directional distance functions. In addition, we explain the relationships between the duality theorem of linear programming and that of homogeneous programming, and show how to apply the latter in those economic models in which linear programming has been utilized.

Variational problems and its duality in banach spaces

The concept of duality for the variational problems is introduced in general Banach spaces. Different forms of duality such as Mangasarian and Mond-Weir type are studied. Mond-Weir type duality is considered to weaken the convexity requirements. Many duality (weak, strong and converse) results are established under generalized convexity assumptions. Again, examples and counterexamples are discussed in support of the investigation.

Optimality conditions and duality results in Banach space under ρ − (η, θ)-B-invexity

In this paper we introduce the notion of \(\rho -(\eta , \theta )\)-B-invex function and generalized \(\rho -(\eta ,\theta )\)-B-invex function between Banach spaces. By considering these functions, sufficient optimality conditions are obtained for a single objective optimization problem in Banach space. Duality results (i.e. weak duality, strong duality and converse duality of Mond–Weir type and similar to Mixed type duals) are established under \(\rho -(\eta , \theta )\)-B-invexity and weak and strong duality of Mond–Weir type dual are also established under generalized \(\rho -(\eta ,\theta )\)-B-invexity in Banach space.

Duality for $A_\infty$ weights on the real line

Under the same bounds on G_q -constants and A_p -constants, the optimal exponents for sharp inclusions between Gehring G_q -class of weights and Muckenhoupt A_p -class ( 1 < p, q < \infty ) are Hölder conjugate, if p and q are conjugate. This is a consequence of a representation theorem of A_\infty weights in terms of W^{1,r} -biSobolev maps and a duality result between G_q and A_p classes in dimension one. We prove also that sharp a priori bounds on constants correspond under the Hölder conjugate mapping \phi(t)= \frac{t}{t-1} .

Second-order optimality and duality in vector optimization over cones

In this paper, we introduce the notion of a second-order cone- convex function involving second-order directional derivative. Also, second-order cone-pseudoconvex, second-order cone-quasiconvex and other related functions are defined. Second-order optimality and Mond-Weir type duality results are derived for a vector optimization problem over conesnusing the introduced classes of functions.

Pollution-Generating Technologies and Disposability Assumptions

This paper exams the concept of Pollution-generating Technologies (PgT). The main goal of this methodological contribution is to reveal any PgT in production processes compatible with a minimal set of assumptions. We model PgT using a congestion approach of the output set relaxing strong disposability assumption. To this end, we define the B-disposal assumption, that is a kind of limited strong disposability. The B-disposal assumption reflects cost disposability assumption with respect to the undesirable outputs. This disposability assumption leads to a new duality result between an output distance function and the revenue function with possibly negative shadow prices. A sample of 13 representative French airports is considered over the period 2007-2011, in order to implement the new B-disposal assumption on convex non-parametric technologies.

Congestion in production correspondences

This contribution aims to detect and measure more severe forms of congestion than the ones that could hitherto be evaluated in axiomatic production theory. To this end, we define a new S-disposal axiom, a kind of limited strong disposability. This S-disposal assumption leads to a duality result between a general input directional distance function and the cost function that is weaker than the ones established in the literature. Finally, we indicate how finite data sets can or cannot be rationalized by a minimal technology compatible with S-disposal, thereby generalizing the nonparametric weak axiom of cost minimization test.

Total shift during the first passages of Markov-modulated Brownian motion with bilateral ph-type jumps: Formulas driven by the minimal solution matrix of a Riccati equation

ABSTRACTThis article describes our study of the total shift during the first passages (one-sided and two-sided exit times) of Markov-modulated Brownian motion with bilateral ph-type jumps, which is referred to as MMBM. The total shift is defined as the value of a so-called shift process at the first passage epochs of the MMBM. The shift process, introduced by Bean and O’Reilly, behaves like a continuous Markovian fluid process; that is, it increases or decreases linearly with slopes regulated by the underlying Markov process that determines the path of the MMBM. Hence, the notion of total shift, which includes the first passage times of the MMBM as special cases, is useful for describing various performance measures of systems modeled by the MMBM. In this article, we present formulas for the Laplace–Stieltjes transform matrices of the total shift during various first passages of the MMBM. In particular, a Riccati equation is derived so that a matrix associated with the Laplace–Stieltjes transform of the total shift during the first return time of the MMBM is its minimal non-negative solution matrix. With this solution matrix, the Laplace–Stieltjes transform matrices can be obtained without much additional work. Furthermore, it is shown that the Riccati equation satisfies the conditions for the Newton scheme to have quadratic convergence, which enables us to use algorithms with quadratic convergence, such as Newton’s method and the Stochastic Doubling Algorithm, to compute the presented matrix-driven formulas. For the analyses, we take an approach based on approximating the MMBM with a sequence of scaled Markov-modulated fluid flows with bilateral ph-type jumps, referred to as MMFF, that weakly converge to the MMBM. Another contribution of this article is that duality results are derived in relation to the MMBM, which is an extension of the duality theorems developed by Ahn and Ramaswami for an MMFF without a jump.

Duality for Ext-groups and extensions of discrete series for graded Hecke algebras

In this paper, we study extensions of graded affine Hecke algebra modules. In particular, based on an explicit projective resolution on graded affine Hecke algebra modules, we prove a duality result for Ext-groups. This duality result with an Ind-Res resolution gives an algebraic proof of the fact that all higher Ext-groups between a tempered module and a discrete series vanish.

A functorial extension of the abelian Reidemeister torsions of three-manifolds

Let \mathbb F be a field and let G \subset \mathbb F \setminus \{0\} be a multiplicative subgroup. We consider the category \mathsf {Cob}_G of 3 -dimensional cobordisms equipped with a representation of their fundamental group in G , and the category \mathsf {Vect}_{\mathbb F, \pm G} of \mathbb F -linear maps defined up to multiplication by an element of \pm G . Using the elementary theory of Reidemeister torsions, we construct a "Reidemeister functor" from \mathsf {Cob}_G to \mathsf {Vect}_{\mathbb F, \pm G} . In particular, when the group G is free abelian and \mathbb F is the field of fractions of the group ring \mathbb Z [G] , we obtain a functorial formulation of an Alexander-type invariant introduced by Lescop for 3 -manifolds with boundary; when G is trivial, the Reidemeister functor specializes to the TQFT developed by Frohman and Nicas to enclose the Alexander polynomial of knots. The study of the Reidemeister functor is carried out for any multiplicative subgroup G\subset \mathbb F \setminus \{0\} . We obtain a duality result and we show that the resulting projective representation of the monoid of homology cobordisms is equivalent to the Magnus representation combined with the relative Reidemeister torsion.

Lower level duality and the global solution of generalized semi-infinite programs

The reformulation of generalized semi-infinite programs (GSIP) to simpler problems is considered. These reformulations are achieved under the assumption that a duality property holds for the lower level program (LLP). Lagrangian duality is used in the general case to establish the relationship between the GSIP and a related semi-infinite program (SIP). Practical aspects of this reformulation, including how to bound the duality multipliers, are also considered. This SIP reformulation result is then combined with recent advances for the global, feasible solution of SIP to develop a global, feasible point method for the solution of GSIP. Reformulations to finite nonlinear programs, and the practical aspects of solving these reformulations globally, are also discussed. When the LLP is a linear program or second-order cone program, specific duality results can be used that lead to stronger results. Numerical examples demonstrate that the global solution of GSIP is computationally practical via the solution of these duality-based reformulations.

A Mathematical Theory for Clustering in Metric Spaces

Clustering is one of the most fundamental problems in data analysis and it has been studied extensively in the literature. Though many clustering algorithms have been proposed, clustering theories that justify the use of these clustering algorithms are still unsatisfactory. In particular, one of the fundamental challenges is to address the following question: What is a cluster in a set of data points? In this paper, we make an attempt to address such a question by considering a set of data points associated with a distance measure (metric). We first propose a new cohesion measure in terms of the distance measure. Using the cohesion measure, we define a cluster as a set of points that are cohesive to themselves. For such a definition, we show there are various equivalent statements that have intuitive explanations. We then consider the second question: How do we find clusters and good partitions of clusters under such a definition? For such a question, we propose a hierarchical agglomerative algorithm and a partitional algorithm. Unlike standard hierarchical agglomerative algorithms, our hierarchical agglomerative algorithm has a specific stopping criterion and it stops with a partition of clusters. Our partitional algorithm, called the K-sets algorithm in the paper, appears to be a new iterative algorithm. Unlike the Lloyd iteration that needs two-step minimization, our K-sets algorithm only takes one-step minimization. One of the most interesting findings of our paper is the duality result between a distance measure and a cohesion measure. Such a duality result leads to a dual K-sets algorithm for clustering a set of data points with a cohesion measure. The dual K-sets algorithm converges in the same way as a sequential version of the classical kernel K-means algorithm. The key difference is that a cohesion measure does not need to be positive semi-definite.

Optimality conditions and duality for nonsmooth minimax programming problems under generalized invexity

In this paper, we consider a class of nonsmooth minimax programming problems in which functions are locally Lipschitz. Sucient optimality conditions are discussed under locally Lipschitz generalized (?,?)-invex functions. Moreover, usual duality results are proved under the said assumptions.

Higher - order duality for multiobjective optimization problems containing support functions over cones

In this paper a new class of higher order (F, ρ, σ)- type I functions over cones are defined for a multiobjective programming problem, which subsumes several known studied classes. A higher order Mond-Weir type dual program is formulated for a non-differentiable multiobjective programming problem over cones where the objective functions and the constraints contain support functions of compact convex sets in Rn. Weak and strong duality results are established by using the above defined functions.

Residues and duality on semi-local two-dimensional adeles

In this note, we establish a duality result under the residue paring between certain two-dimensional adelic spaces, which are associated to a closed point on an arithmetic surface.

Optimality and duality results for a nondifferentiable multiobjective fractional programming problem

The purpose of this paper is to study a class of nondifferentiable multiobjective fractional programming problems in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we introduce the definition of higher-order $(C,\alpha,\gamma,\rho,d)$ -convex function. A nontrivial example is also constructed which is in this class but not $(F,\alpha,\gamma,\rho,d)$ -convex. Based on the $(C,\alpha,\gamma,\rho,d)$ -convexity, sufficient optimality conditions for an efficient solution of the nondifferentiable multiobjective fractional programming problem are established. Further, a higher-order Mond-Weir type dual is formulated for this problem and appropriate duality results are proved under higher-order $(C,\alpha,\gamma,\rho,d)$ -assumptions.

On optimality conditions and duality results in a class of nonconvex quasidifferentiable optimization problems

In the paper, the class of nonconvex nonsmooth optimization problems with the quasidifferentiable functions is considered. Further, a new notion of nonsmooth generalized convexity, namely, the concept of r-invexity with respect to a convex compact set is introduced. Several conditions for quasidifferentiable r-invexity with respect to a convex compact set are given. Furthermore, the sufficient optimality conditions and several Mond–Weir duality results are established for the considered nonconvex quasidifferentiable optimization problem under assumption that the functions constituting it are r-invex with respect to the same function eta and with respect to convex compact sets which are equal to Minkowski sum of their subdifferentials and superdifferentials. It is also illustrated that, for such nonsmooth extremum problems, the Lagrange multipliers may not be constant.