Fenchel-Rockafellar Duality Research Articles

On the foundations of the maximum entropy principle using Fenchel duality for Shannon and Tsallis entropies

In this work, we address two main objectives. The first one is to provide a rigorous foundation to the maximum entropy principle in statistical physics, by making use of the Fenchel-Rockafellar duality. The second objective is to discuss the well-foundedness of the so-called escort distributions in the context of non-extensive entropy maximization. The duality treatment of maximum entropy confirms the non-rigorous results obtained via the usual variational calculus, however, the use of escort distributions yields undefined behavior when used consistently, and only leads to the desired results when used in an ad-hoc manner.

Augmented Lagrangian methods for degenerate Hamilton–Jacobi equations

We suggest a new approach to solve a class of degenerate Hamilton–Jacobi equations without any assumptions on the emptiness of the Aubry set. It is based on the characterization of the maximal subsolution by means of the Fenchel–Rockafellar duality. This approach enables us to use augmented Lagrangian methods as alternatives to the commonly used methods for numerical approximation of the solution, based on finite difference approximation or on optimal control interpretation of the solution.

Generalized support vector regression: Duality and tensor-kernel representation

In this paper, we study the variational problem associated to support vector regression in Banach function spaces. Using the Fenchel–Rockafellar duality theory, we give an explicit formulation of the dual problem as well as of the related optimality conditions. Moreover, we provide a new computational framework for solving the problem which relies on a tensor-kernel representation. This analysis overcomes the typical difficulties connected to learning in Banach spaces. We finally present a large class of tensor-kernels to which our theory fully applies: power series tensor kernels. This type of kernels describes Banach spaces of analytic functions and includes generalizations of the exponential and polynomial kernels as well as, in the complex case, generalizations of the Szegö and Bergman kernels.

Total variation-based dense depth from multicamera array

Multi-Camera arrays are increasingly employed in both consumer and industrial applications, and various passive techniques are documented to estimate depth from such camera arrays. Current depth estimation methods provide useful estimations of depth in an imaged scene but are often impractical due to significant computational requirements. This paper presents a novel framework that generates a high-quality continuous depth map from multi-camera array/light field cameras. The proposed framework utilizes analysis of the local Epipolar Plane Image (EPI) to initiate the depth estimation process. The estimated depth map is then processed using Total Variation (TV) minimization based on the Fenchel-Rockafellar duality. Evaluation of this method based on a well-known benchmark indicates that the proposed framework performs well in terms of accuracy when compared to the top-ranked depth estimation methods and a baseline algorithm. The test dataset includes both photorealistic and non-photorealistic scenes. Notably, the computational requirements required to achieve an equivalent accuracy are significantly reduced when compared to the top algorithms. As a consequence, the proposed framework is suitable for deployment in consumer and industrial applications.

On Closedness Conditions, Strong Separation, and Convex Duality

In the paper, we describe various applications of closedness and duality theorems from previous works of the author. First, the strong separability of a polyhedron and a linear image of a convex set is characterized. Then, it is shown how stability conditions (known from the generalized Fenchel-Rockafellar duality theory) can be reformulated as closedness conditions. Finally, we present a generalized Lagrangian duality theorem for Lagrangian programs described with cone-convex/cone-polyhedral mappings.

A Fenchel–Rockafellar duality theorem for set-valued optimization

A duality theorem of the Fenchel–Rockafellar type for set-valued optimization problems is presented along with a result for the conjugate of the sum of two set-valued functions and a chain rule. The underlying solution concepts rely on order complete lattices of sets defined via set relations. Set-valued replacements for linear operators are used as dual variables, for example in order to define Fenchel conjugates for set-valued functions. An application to mathematical finance is given.

Controllability of a Class of Newtonian Filtration Equations with Control and State Constraints

This paper addresses a study of the controllability of a class of Newtonian filtration equations, with nonnegative constraints on the control and state variables. When the control enters the system through the whole domain where the equation evolves, we characterize the set of nonnegative targets which are approximately controllable at any time $T>0$. The proof combines the Fenchel–Rockafellar duality theory and a fixed point argument. When the control is restricted to be active in a proper open subset of the whole domain, we prove a negative controllability result by means of a localization technique which reflects the underlying obstruction phenomenon in the system.

A Convex-Analytical Approach to Extension Results for n-Cyclically Monotone Operators

Results concerning extensions of monotone operators have a long history dating back to a classical paper by Debrunner and Flor from 1964. In 1999, Voisei obtained refinements of Debrunner and Flor’s work for n-cyclically monotone operators. His proofs rely on von Neumann’s minimax theorem as well as Kakutani’s fixed point theorem. In this note, we provide a new proof of the central case of Voisei’s work. This proof is more elementary and rooted in convex analysis. It utilizes only Fitzpatrick functions and Fenchel–Rockafellar duality.

Necessary and sufficient conditions for stable conjugate duality

The conjugate duality, which states that inf x ∈ X φ ( x , 0 ) = max v ∈ Y ′ - φ * ( 0 , v ) , whenever a regularity condition on φ is satisfied, is a key result in convex analysis and optimization, where φ : X × Y → R ∪ { + ∞ } is a convex function, X and Y are Banach spaces, Y ′ is the continuous dual space of Y and φ * is the Fenchel–Moreau conjugate of φ . In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, inf x ∈ X { φ ( x , 0 ) + x * ( x ) } = max v ∈ Y ′ { - φ * ( - x * , v ) } , ∀ x * ∈ X ′ and then obtain a new epigraph regularity condition for the conjugate duality. The regularity condition is shown to be much more general than the popularly known interior-point type conditions. As an easy consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel–Rockafellar duality theorem. In the case where one of the functions involved is a polyhedral convex function, we provide generalized interior-point conditions for the epigraph regularity condition. Moreover, we show that a stable Fenchel's duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds.

Approximate controllability and homogenization of a semilinear elliptic problem

The L 2- and H 1-approximate controllability and homogenization of a semilinear elliptic boundary-value problem is studied in this paper. The principal term of the state equation has rapidly oscillating coefficients and the control region is locally distributed. The observation region is a subset of codimension 1 in the case of L 2-approximate controllability or is locally distributed in the case of H 1-approximate controllability. By using the classical Fenchel–Rockafellar's duality theory, the existence of an approximate control of minimal norm is established by means of a fixed point argument. We consider its asymptotic behavior as the rapidly oscillating coefficients H-converge. We prove its convergence to an approximate control of minimal norm for the homogenized problem.

Multiobjective duality for convex ratios

In this paper we present a duality approach for a multiobjective fractional programming problem. The components of the vector objective function are particular ratios involving the square of a convex function and a positive concave function. Applying the Fenchel–Rockafellar duality theory for a scalar optimization problem associated to the multiobjective primal, a dual problem is derived. This scalar dual problem is formulated in terms of conjugate functions and its structure gives an idea about how to construct a multiobjective dual problem in a natural way. Weak and strong duality assertions are presented.

A comparison of constraint qualifications in infinite-dimensional convex programming revisited

In 1990 Gowda and Teboulle published the paper [16], making a comparison of several conditions ensuring the Fenchel-Rockafellar duality formulainf{f(x) + g(Ax) | x ∈ X} = max{−f*(A*y*) − g*(− y*) | y* ∈ Y*}.Probably the first comparison of different constraint qualification conditions was made by Hiriart-Urruty [17] in connection with ε-subdifferential calculus. Among them appears, as the basic sufficient condition, the formula for the conjugate of the corresponding function; such functions are: f1 + f2, g o A, max{fl,…, fn}, etc. In fact strong duality formulae (like the one above) and good formulae for conjugates are equivalent and they can be used to obtain formulae for ε-subdifferentials, using a technique developed in [17] and extensively used in [46].

A decomposition-dualization approach for solving constrained convex minimization problems with applications to discretized obstacle problems

In this paper, we shall be concerned with the solution of constrained convex minimization problems. The constrained convex minimization problems are proposed to be transformable into a convex-additively decomposed and almost separable form, e.g. by decomposition of the objective functional and the restrictions. Unconstrained dual problems are generated by using Fenchel-Rockafellar duality. This decomposition-dualization concept has the advantage that the conjugate functionals occuring in the derived dual problem are easily computable. Moreover, the minimum point of the primal constrained convex minimization problem can be obtained from any maximum point of the corresponding dual unconstrained concave problem via explicit return-formulas. In quadratic programming the decomposition-dualization approach considered here becomes applicable if the quadratic part of the objective functional is generated byH-matrices. Numerical tests for solving obstacle problems in ?1 discretized by using piecewise quadratic finite elements and in ?2 by using the five-point difference approximation are presented.

A Minimax Theorem

A new, general criterion is given for ensuring that a closed saddle function has a nonempty compact set of saddlepoints. Under this criterion it is shown also that every minimaximizing sequence clusters around some saddlepoint. A comparable theorem is given for semicontinuous quasi-saddle functions. The new criterion is applied to constrained saddlepoint problems and to the Fenchel-Rockafellar duality model for constrained convex minimization. Finally, the relationship to existing saddlepoint results is explored in detail.

Dual analysis of frictionless problems by displacement and equilibrium finite elements

The paper deals with the duality of the frictionless contact problem in the sense of convex analysis, contact between an elastic body and a rigid foundation under small deformation conditions. A new method using equilibrium finite elements is proposed. Virtual work theorems are first shown under variational equality forms. Fenchel-Rockafellar duality and Arrow-Hurwicz duality are used to deduce four variational principles of elastic bodies in contact. The boundary contact finite elements are then formulated from these theoretical foundations of convex analysis. A new interpretation of equilibrium finite elements and their associated boundary contact finite elements is described. Lastly, the different contact finite elements, their properties and numerical results are described. Agreement with analytical solutions is satisfactory.

Some new applications of the Fenchel-Rockafellar duality theorem: Lagrange multiplier theorems and hyperplane theorems for convex optimization and best approximation

XEM xsM (the extension to complex scalars can be obtained with the usual methods-see [3,4]) and therefore, from now on, we shall omit the adjective “real”. In several works it has been observed that Theorem A yields, as a particular case, the value of the intimum of a continuous convex functional f on a convex subset G of E, by taking in (1.1) h = -6,, where 6, is the indicator of the set G, i.e.,

Symmetric Duality for Structured Convex Programs

A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar's general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.

Symmetric duality for structured convex programs

A fully symmetric duality model is presented which subsumes the classical treatments given by Duffin (1956), Eisenberg (1961) and Cottle (1963) for linear, homogeneous and quadratic convex programming. Moreover, a wide variety of other special objective functional structures, including homogeneity of any nonzero degree, is handled with equal ease. The model is valid in spaces of arbitrary dimension and treats explicitly systems of both nonnegativity and linear inequality constraints, where the partial orderings may correspond to nonpolyhedral convex cones. The approach is based on augmenting the Fenchel-Rockafellar duality model (1951, 1967) with cone structure to handle constraint systems of the type mentioned. The many results and insights from Rockafellar’s general perturbational duality theory can thus be brought to bear, particularly on sensitivity analysis and the interpretation of dual variables. Considerable attention is devoted to analysis of suboptimizations occurring in the model, and the model is shown to be the projection of another model.

Applications of the Hahn-Banach Theorem in Approximation Theory

Applications of the Hahn-Banach Theorem in Approximation Theory