Fenchel Duality Theorem Research Articles

Fenchel Duality Based Dictionary Learning for Restoration of Noisy Images

Dictionary learning based sparse modeling has been increasingly recognized as providing high performance in the restoration of noisy images. Although a number of dictionary learning algorithms have been developed, most of them attack this learning problem in its primal form, with little effort being devoted to exploring the advantage of solving this problem in a dual space. In this paper, a novel Fenchel duality based dictionary learning (FD-DL) algorithm has been proposed for the restoration of noise-corrupted images. With the restricted attention to the additive white Gaussian noise, the sparse image representation is formulated as an 2-1 minimization problem, whose dual formulation is constructed using a generalization of Fenchel’s duality theorem and solved under the augmented Lagrangian framework. The proposed algorithm has been compared with four state-of-the-art algorithms, including the local pixel grouping-principal component analysis, method of optimal directions, K-singular value decomposition, and beta process factor analysis, on grayscale natural images. Our results demonstrate that the FD-DL algorithm can effectively improve the image quality and its noisy image restoration ability is comparable or even superior to the abilities of the other four widely-used algorithms.

COHERENT AND CONVEX HEDGING ON ORLICZ HEARTS IN INCOMPLETE MARKETS

Every contingent claim is unable to be replicated in the incomplete markets. Shortfall risk is considered with some risk exposure. We show how the dynamic optimization problem with the capital constraint can be reduced to the problem to find an optimal modified claim where is a randomized test in the static problem. Convex and coherent risk measures defined in the Orlicz hearts spaces, , are used as risk measure. It can be shown that we have the same results as in [21, 22] even though convex and coherent risk measures defined in the Orlicz hearts spaces, , are used. In this paper, we use Fenchel duality Theorem in the literature to deduce necessary and sufficient optimality conditions for the static optimization problem using convex duality methods.

Sandwich theorem for quasiconvex functions and its applications

In convex programming, sandwich theorem is very important because it is equivalent to Fenchel duality theorem. In this paper, we investigate a sandwich theorem for quasiconvex functions. Also, we consider some applications for quasiconvex programming.

Fenchel’s Duality Theorem for Nearly Convex Functions

We present an extension of Fenchel's duality theorem by weakening the convexity assumptions to near convexity. These weak hypotheses are automatically fulfilled in the convex case. Moreover, we show by a counterexample that a further extension to closely convex functions is not possible under these hypotheses.

The Fenchel duality, S-procedure, and the Yakubovich-Kalman lemma

The role of the Fenchel duality in the theorem on the losslessness of the S-procedure and in Yakubovich-Kalman lemma is studied. The Fenchel duality theorem implying the well-known results on the losslessness of the S-procedure is formulated. A relation between the Yakubovich-Kalman lemma and a special extremal problem defined on a set of positive-demidefinite solutions of the generalized Lyapunov inclusion is derived. Every assertion of the lemma is shown to be necessary and sufficient for the value of the extremal problem to be bounded.

Generalized Newton methods for the 2D-Signorini contact problem with friction in function space

The 2D-Signorini contact problem with Tresca and Coulomb friction is discussed in infinite-dimensional Hilbert spaces. First, the problem with given friction (Tresca friction) is considered. It leads to a constraint non-differentiable minimization problem. By means of the Fenchel duality theorem this problem can be transformed into a constrained minimization involving a smooth functional. A regularization technique for the dual problem motivated by augmented Lagrangians allows to apply an infinite-dimensional semi-smooth Newton method for the solution of the problem with given friction. The resulting algorithm is locally superlinearly convergent and can be interpreted as active set strategy. Combining the method with an augmented Lagrangian method leads to convergence of the iterates to the solution of the original problem. Comprehensive numerical tests discuss, among others, the dependence of the algorithm's performance on material and regularization parameters and on the mesh. The remarkable efficiency of the method carries over to the Signorini problem with Coulomb friction by means of fixed point ideas.

Fenchel duality, Fitzpatrick functions and the Kirszbraun–Valentine extension theorem

We present a new proof of the classical Kirszbraun-Valentine extension theorem. Our proof is based on the Fenchel duality theorem from convex analysis and an analog for nonexpansive mappings of the Fitzpatrick function from monotone operator theory.

Fenchel duality theorem in multiobjective programming problems with set functions

In this paper, we characterize a vector-valued convex set function by its epigraph. The concepts of a vector-valued set function and a vector-valued concave set function are given respectively. The definitions of the conjugate functions for a vector-valued convex set function and a vector-valued concave set function are introduced. Then a Fenchel duality theorem in multiobjective programming problem with set functions is derived.

Proof of a Conjecture by Deutsch, Li, and Swetits on Duality of Optimization Problems

A conjecture of Deutsch, Li, and Swetits on duality relationships among three optimization problems is shown to hold true. The proof relies on a reformulation of one of the problems in a suitable product space, to which then a version of the classical Fenchel duality theorem applies.

A class of extremum problems related to agency models with imperfect monitoring

The cost minimization problem in an agency model with imperfect monitoring is considered. Under the first order approach, this can be stated as a convex minimization problem with linear inequality and equality constraints in a generally infinite dimensional function space. We apply the Fenchel Duality Theorem, and obtain as a dual problem a concave maximization problem of finite dimension. In particular, a Lagrange multiplier description of the optimal solution to the cost minimization problem is derived, justifying and extending thus the approach of Kim (1995). By the duality, the dependence of the minimum cost value on the information system used becomes particularly visible. The minimum cost value behaves monotonically w.r.t. the convex ordering of certain distributions induced by the competing information systems. Under the standard inequality constraint, one is led to the distributions of the score functions of the information systems and their convex order relation. It is shown that also for multivariate actions, Blackwell sufficiency implies the convex order relation of the score function distributions. A further result refers to a multi-agents model recently considered by Budde (1997), when the maximum of n independent and identically distributed (i.i.d.) univariate output variables is focussed. If two univariate information systems have monotone likelihood ratios, then the convex ordering between the two score function distributions implies the weaker convex increasing ordering between the distributions of the same score functions under the maximum distributions.

Sum theorems for monotone operators and convex functions

In this paper, we derive sufficient conditions for the sum of two or more maximal monotone operators on a reflexive Banach space to be maximal monotone, and we achieve this without any renorming theorems or fixed-point-related concepts. In the course of this, we will develop a generalization of the uniform boundedness theorem for (possibly nonreflexive) Banach spaces. We will apply this to obtain the Fenchel Duality Theorem for the sum of two or more proper, convex lower semicontinuous functions under the appropriate constraint qualifications, and also to obtain additional results on the relation between the effective domains of such functions and the domains of their subdifferentials. The other main tool that we use is a standard minimax theorem.

Semi-infinite programming duality for order restricted statistical inference models

The equivalence of multinomial maximum likelihood and the isotonic projection problem: $$\inf \left\{ {\sum\limits_{i = 1}^n {p_i \ln (p_i /r_i )|p \in K,a convex} {\mathbf{ }}subset of the probability vectors{\mathbf{ }}p} \right\}$$ can be established using Fenchel's Duality Theorem and subgradient and complementary slackness relationships of convex analysis, all taking place over the real numbers.

Duality and infinite dimensional optimization

The purpose is to, establish various duality theorems, including a generalization of the Fenchel duality theorem of Attouch and Brezis to convex processes under relaxed interiority conditions and show the usefulness of the Attouch and Brezis type conditions in optimization. The significance of the interiority conditions is that they provide easily verifiable conditions for a large class of infinite dimensional convex programs; whereas the standard Slater constraint qualification fails. The results are applied to obtain duality theorems for general systems of inequalities and minimax results

The Fenchel duality theorem in Fréchet spaees

We generalize the Fenchel duality and inf-convolution theorems from convex optimization to Frechet spaces. Our approach is simple and direct, and is to first prove a more general theorem in which we consider only one function and replace the product of spaces by an arbitrary Frechet space. These results extend those obtained recently by Attouch and Brezis. Applications of our Generalizations are given.

Nonparametric maximum likelihood estimation of survival functions with a general stochastic ordering and its dual

This paper discusses estimation of survival functions under an arbitrary partial stochastic ordering of the underlying populations. This technique is especially useful when data are not easily acquired from one or more of the populations, but data from the other populations are available. Estimation of the ordered survival functions entails use of a Fenchel duality theorem that results in an algorithm to identify and resolve violations of the underlying ordering. The algorithm produces estimates which converge to the maximum likelihood estimates of the survivalfunctions given the underlying ordering. The method is illustrated by an example.

Duality ofIProjections and Maximum Likelihood Estimation for Log-Linear Models Under Cone Constraints

Abstract Important order restrictions and general log-linear models for multinomial experiments can often be expressed by requiring that the vector composed of the logs of the probabilities fall within a closed convex cone. Here we exhibit a duality relationship (by way of a Fenchel duality theorem) between these types of problems and projections in I-divergence geometry. We show that many cone-constrained maximum likelihood estimation problems are exactly equivalent to an I projection onto a translation of the negative polar cone. This duality relationship permits the concise characterization of cone-restricted maximum likelihood estimates and the use of iterative algorithms proposed by Csiszar (1975) and Dykstra (1985b), which are analogous to the iterative proportional-fitting procedure. Computational aspects of maximum likelihood estimates are discussed, and a relationship with least squares projections onto isotonic cones is presented. We discuss examples in detail to exhibit the usefulness and power...

Epigraphs of convex set functions

In this paper, we characterize a convex set function by its epigraph. The w ∗-semicontinuities for set functions are defined. We identify a convex set function with a functional in L ∞ and show that the w ∗-closure of such a functional is a convex functional. A Fenchel duality theorem for set functions is then derived.

Duality in mathematical programming of set functions: On Fenchel duality theorem

The optimization problem of a set function defined on a family a′ of measurable subsets in an atomless finite measure space ( X, a, m) is investigated. The generalized Fenchel theorem is formulated and proved in this note.

Equivalence between a generalized fenchel duality theorem and a saddle-point theorem for fractional programs

The convex case of a fractional program is considered. By the aid of a duality theory for mathematical programming involving a maximum and minimum operator, and by defining new operators, we obtain three equivalent duality theorems for three pairs of primal-dual programs. The second and the third one are the saddle-point theorem and a generalized Fenchel duality theorem, respectively.

A geometrical interpretation of ideal body problems

Summary. For linear geophysical inverse problems, the exercise of finding a greatest lower bound on the uniform norms of positive solutions fitting N data, is shown to have a geometrical counterpart in the N-dimensional space of N-tuples of real numbers. By application of the Fenchel Duality Theorem, we demonstrate that the problem is equivalent to the discovery of a particular hyperplane tangent to a convex set in this space. As examples in the case of two data, the new formulation is applied to the problems of recovering density information from planetary mass and moment of inertia, and from two vertical gravity anomalies.