A comparison of alternative c-conjugate dual problems in infinite convex optimization

In this work, we obtain a Fenchel–Lagrange dual problem for an infinite dimensional optimization primal one, via perturbational approach and using a conjugation scheme called c-conjugation instead of classical Fenchel conjugation. This scheme is based on the generalized convex conjugation theory. We analyse some inequalities between the optimal values of Fenchel, Lagrange and Fenchel–Lagrange dual problems and we establish sufficient conditions under which they are equal. Examples where such inequalities are strictly fulfilled are provided. Finally, we study the relations between the optimal solutions and the solvability of the three mentioned dual problems.     

Stable strong Fenchel and Lagrange duality for evenly convex optimization problems

By means of a conjugation scheme based on generalized convex conjugation theory instead of Fenchel conjugation, we build an alternative dual problem, using the perturbational approach, for a general optimization one defined on a separated locally convex topological space. Conditions guaranteeing strong duality for primal problems which are perturbed by continuous linear functionals and their respective dual problems, which is named stable strong duality, are established. In these conditions, the fact that the perturbation function is evenly convex will play a fundamental role. Stable strong duality will also be studied in particular for Fenchel and Lagrange primal–dual problems, obtaining a characterization for Fenchel case.     

Fenchel-duality and separably-infinite programs

In two recent papers Chabnes, Gbibie, and Kortanek [3], [4] studied a special class of infinite linear programs where only a finite number of variables appear in an infinite number of constraints and where only a finite number of constraints have an infinite number of variables. Termed separably-infinite programs, their duality was used to characterize a class of saddle value problems as a uniextremai principle.

We show how this characterization can be derived and extended within Fenchel and Rockafellar duality, and that the values of the dual separably-infinite programs embrace the values of the Fenchel dual pair within their interval. The development demonstrates that the general finite dimensional Fenchel dual pair is equivalent to a dual pair of separably-infinite programs when certain cones of coefficients are closed.     

Symbolic Fenchel Conjugation

Of key importance in convex analysis and optimization is the notion of duality, and in particular that of Fenchel duality. This work explores improvements to existing algorithms for the symbolic calculation of subdifferentials and Fenchel conjugates of convex functions defined on the real line. More importantly, these algorithms are extended to enable the symbolic calculation of Fenchel conjugates on a class of real-valued functions defined on $\mathbb{R}^n$ . These algorithms are realized in the form of the Maple package SCAT.     

Stable and Total Fenchel Duality for Composed Convex Optimization Problems

In this paper, we consider the composed convex optimization problem which consists in minimizing the sum of a convex function and a convex composite function. By using the properties of the epigraph of the conjugate functions and the subdifferentials of convex functions, we give some new constraint qualifications which completely characterize the strong Fenchel duality and the total Fenchel duality for composed convex optimiztion problem in real locally convex Hausdorff topological vector spaces.     

Note on the Fenchel transform in the Heisenberg group

Given a real-valued function defined on the Heisenberg group H, we provide a definition of abstract convexity and Fenchel transform in H, that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the classical case, a convex function can be characterized via its iterated Fenchel transform; the properties of the H-subdifferential play a crucial role.     

Legendre–Fenchel duality in elasticity

We show that the displacement and strain formulations of the displacement–traction problem of three-dimensional linearized elasticity can be viewed as Legendre–Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre–Fenchel duality.     

Scalar representation and conjugation of set-valued functions

We consider functions with values in the power set of a pre-ordered, separated locally convex space with closed convex images. To each such function, a family of scalarizations is given which completely characterizes the original function. A concept of a Legendre–Fenchel conjugate for set-valued functions is introduced and identified with the conjugates of the scalarizations. To the set-valued conjugate, a full calculus is provided, including a biconjugation theorem, a chain rule and weak and strong duality results of the Fenchel–Rockafellar type.     

A generalization of Fenchel duality theory

This paper deals with an extention of Fenchel duality theory to fractional extremum problems, i.e., problems having a fractional objective function. The main result is obtained by regarding the classic Fenchel theorem as a decomposition property for the extremum of a sum of functions into a sum of extrema of functions, and then by extending it to the case where the addition is replaced by the quotient. This leads to a generalization of the classic concept of conjugate function. Several remarks are made about the conceivable further generalizations to other kinds of decomposition.     

A new Fenchel dual problem in vector optimization

We introduce a new Fenchel dual for vector optimization problems inspired by the form of the Fenchel dual attached to the scalarized primal multiobjective problem. For the vector primal-dual pair we prove weak and strong duality. Furthermore, we recall two other Fenchel-type dual problems introduced in the past in the literature, in the vector case, and make a comparison among all three duals. Moreover, we show that their sets of maximal elements are equal.     

New Formulas for the Fenchel Subdifferential of the Conjugate Function

Following (López and Volle, J Convex Anal 17, 2010) we provide new formulas for the Fenchel subdifferential of the conjugate of functions defined on locally convex spaces. In particular, this allows deriving expressions for the minimizers set of the lower semicontinuous convex hull of such functions. These formulas are written by means of primal objects related to the subdifferential of the initial function, namely a new enlargement of the Fenchel subdifferential operator.     

A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.      

Solving capacitated facility location problems by Fenchel cutting planes

In this paper, we apply the Fenchel cutting planes methodology to Capacitated Facility Location problems. We select a suitable knapsack structure from which depth cuts can be obtained. Moreover, we simultaneously obtain a primal heuristic solution. The lower and upper bounds achieved by our procedure are compared with those provided by Lagrangean relaxation of the demand constraints. As the computational results show the Fenchel cutting planes methodology outperforms the Lagrangean one, both in the obtaining of the bounds and in the effectiveness of the branch and bound algorithm using each relaxation as the initial formulation.     

Extended Farkas’s Lemmas and Strong Dualities for Conic Programming Involving Composite Functions

The paper is devoted to the study of a new class of conic constrained optimization problems with objectives given as differences of a composite function and a convex function. We first introduce some new notions of constraint qualifications in terms of the epigraphs of the conjugates of these functions. Under the new constraint qualifications, we provide necessary and sufficient conditions for several versions of Farkas lemmas to hold. Similarly, we provide characterizations for conic constrained optimization problems to have the strong or stable strong dualities such as Lagrange, Fenchel–Lagrange or Toland–Fenchel–Lagrange duality.     

Some characterizations of duality for DC optimization with composite functions

In this paper, by virtue of the epigraph technique, we first introduce some new regularity conditions and then obtain some complete characterizations of the Fenchel–Lagrange duality and the stable Fenchel–Lagrange duality for a new class of DC optimization involving a composite function. Moreover, we apply the strong and stable strong duality results to obtain some extended (stable) Farkas lemmas and (stable) alternative type theorems for this DC optimization problem. As applications, we obtain the corresponding results for a composed convex optimization problem, a DC optimization problem, and a convex optimization problem with a linear operator, respectively.     

A stability estimate for the Aleksandrov–Fenchel inequality under regularity assumptions

We give, under appropriate regularity assumptions, a strengthening of the Aleksandrov–Fenchel inequality in the form of a stability estimate.     

Young Duality and Aggregation of Balances

Doklady Mathematics - An operation generalizing convolution is introduced using the Young transform and Fenchel’s duality theorem. Based on this operation, an aggregation procedure for a...     

Faster than the Fast Legendre Transform,the Linear-time Legendre Transform

A new algorithm to compute the Legendre–Fenchel transform is proposed and investigated. The so-called Linear-time Legendre Transform (LLT) improves all previously known Fast Legendre Transform algorithms by reducing their log-linear worst-case time complexity to linear. Since the algorithm amounts to computing several convex hulls and sorting, any convex hull algorithm well-suited for a particular problem gives a corresponding LLT algorithm. After justifying the convergence of the Discrete Legendre Transform to the Legendre–Fenchel transform, an extended computation time complexity analysis is given and confirmed by numerical tests. Finally, the LLT is illustrated with several examples and a LLT MATLAB package is described. This revised version was published online in June 2006 with corrections to the Cover Date.     

Regularity conditions via generalized interiority notions in convex optimization: New achievements and their relation to some classical statements

For the existence of strong duality in convex optimization regularity conditions play an indisputable role. In this article we mainly deal with regularity conditions formulated by means of different generalizations of the notion of interior of a set. The primal–dual pair we investigate is a general one expressed in the language of a perturbation function and by employing its Fenchel–Moreau conjugate. After providing an overview on the generalized interior-point conditions that exist in the literature we introduce several new ones formulated by means of the quasi-interior and quasi-relative interior. We underline the advantages of the new conditions vis-á-vis the classical ones and illustrate our investigations by numerous examples. We conclude this article by particularizing the general approach to the classical Fenchel and Lagrange duality concepts.     

A Cut and Branch Approach for the Capacitated <Emphasis Type="Italic">p</Emphasis>-Median Problem Based on Fenchel Cutting Planes

The capacitated p-median problem (CPMP) consists of finding p nodes (the median nodes) minimizing the total distance to the other nodes of the graph, with the constraint that the total demand of the nodes assigned to each median does not exceed its given capacity. In this paper we propose a cutting plane algorithm, based on Fenchel cuts, which allows us to considerably reduce the integrality gap of hard CPMP instances. The formulation strengthened with Fenchel cuts is solved by a commercial MIP solver. Computational results show that this approach is effective in solving hard instances or considerably reducing their integrality gap.