Conjugate Duality Theory Research Articles

Perturbation of Image and conjugate duality for vector optimization

<p style='text-indent:20px;'>This paper aims at employing the image space approach to investigate the conjugate duality theory for general constrained vector optimization problems. We introduce the concepts of conjugate map and subdifferential by using two types of maximums. We also construct the conjugate duality problems via a perturbation method. Moreover, the separation condition is proposed by means of vector weak separation functions. Then, it is proved to be a new sufficient condition, which ensures the strong duality theorem. This separation condition is different from the classical regular conditions in the literature. Simultaneously, the application to a nonconvex multi-objective optimization problem is shown to verify our main results.</p>

Regularity conditions and Farkas-type results for systems with fractional functions

This paper deals with some new versions of Farkas-type results for a system involving cone convex constraint, a geometrical constraint as well as a fractional function. We first introduce some new notions of regularity conditions in terms of the epigraphs of the conjugate functions. By using these regularity conditions, we obtain some new Farkas-type results for this system using an approach based on the theory of conjugate duality for convex or DC optimization problems. Moreover, we also show that some recently obtained results in the literature can be rediscovered as special cases of our main results.

A convex duality approach for pricing contingent claims under partial information and short selling constraints

ABSTRACTWe consider the pricing problem facing a seller of a contingent claim. We assume that this seller has some general level of partial information, and that he is not allowed to sell short in certain assets. This pricing problem, which is our primal problem, is a constrained stochastic optimization problem. We derive a dual to this problem by using the conjugate duality theory introduced by Rockafellar. Furthermore, we give conditions for strong duality to hold. This gives a characterization of the price of the claim involving martingale- and super-martingale conditions on the optional projection of the price processes.

Risk preferences on the space of quantile functions

We propose a novel approach to quantification of risk preferences on the space of nondecreasing functions. When applied to law invariant risk preferences among random variables, it compares their quantile functions. The axioms on quantile functions impose relations among comonotonic random variables. We infer the existence of a numerical representation of the preference relation in the form of a quantile-based measure of risk. Using conjugate duality theory by pairing the Banach space of bounded functions with the space of finitely additive measures on a suitable algebra $$\varSigma $$ Σ , we develop a variational representation of the quantile-based measures of risk. Furthermore, we introduce a notion of risk aversion based on quantile functions, which enables us to derive an analogue of Kusuoka representation of coherent law-invariant measures of risk.

Continuity concepts for set-valued functions and a fundamental duality formula for set-valued optimization

Over the past few years a theory of conjugate duality for set-valued functions that map into the set of upper closed subsets of a preordered topological vector space has been developed. For scalar duality theory, continuity of convex functions plays an important role. For set-valued maps, different notions of continuity exist. We will compare the most prevalent ones for the special case where the image space is the set of upper closed subsets of a preordered topological vector space and analyze which of the results can be conveyed from the extended real-valued case.Moreover, we present a fundamental duality formula for set-valued optimization, using the weakest of the continuity concepts under consideration for a regularity condition.

Looking for appropriate qualification conditions for subdifferential formulae and dual representations for convex risk measures

A fruitful idea, when providing subdifferential formulae and dual representations for convex risk measures, is to make use of the conjugate duality theory in convex optimization. In this paper we underline the outstanding role played by the qualification conditions in the context of different problem formulations in this area. We show that not only the meanwhile classical generalized interiority point conditions come here to bear, but also a recently introduced one formulated by means of the quasi-relative interior.

Optimization problems in statistical learning: Duality and optimality conditions

Regularization methods are techniques for learning functions from given data. We consider regularization problems the objective function of which consisting of a cost function and a regularization term with the aim of selecting a prediction function f with a finite representation f ( · ) = ∑ i = 1 n c i k ( · , X i ) which minimizes the error of prediction. Here the role of the regularizer is to avoid overfitting. In general these are convex optimization problems with not necessarily differentiable objective functions. Thus in order to provide optimality conditions for this class of problems one needs to appeal on some specific techniques from the convex analysis. In this paper we provide a general approach for deriving necessary and sufficient optimality conditions for the regularized problem via the so-called conjugate duality theory. Afterwards we employ the obtained results to the Support Vector Machines problem and Support Vector Regression problem formulated for different cost functions.

Robust conjugate duality for convex optimization under uncertainty with application to data classification

In this paper we present a robust conjugate duality theory for convex programming problems in the face of data uncertainty within the framework of robust optimization, extending the powerful conjugate duality technique. We first establish robust strong duality between an uncertain primal parameterized convex programming model problem and its uncertain conjugate dual by proving strong duality between the deterministic robust counterpart of the primal model and the optimistic counterpart of its dual problem under a regularity condition. This regularity condition is not only sufficient for robust duality but also necessary for it whenever robust duality holds for every linear perturbation of the objective function of the primal model problem. More importantly, we show that robust strong duality always holds for partially finite convex programming problems under scenario data uncertainty and that the optimistic counterpart of the dual is a tractable finite dimensional problem. As an application, we also derive a robust conjugate duality theorem for support vector machines which are a class of important convex optimization models for classifying two labelled data sets. The support vector machine has emerged as a powerful modelling tool for machine learning problems of data classification that arise in many areas of application in information and computer sciences.

Dual Representations for Convex Risk Measures via Conjugate Duality

The aim of this paper is to give dual representations for different convex risk measures by employing their conjugate functions. To establish the formulas for the conjugates, we use on the one hand some classical results from convex analysis and on the other hand some tools from the conjugate duality theory. Some characterizations of so-called deviation measures recently given in the literature turn out to be direct consequences of our results.

Existence results and gap functions for the generalized equilibrium problem with composed functions

In this paper we provide first existence results for solutions of the generalized equilibrium problem with composed functions ( G E P C ) under generalized convexity assumptions. Then we construct by employing some tools specific to the theory of conjugate duality two gap functions for ( G E P C ) . The importance of these gap functions is to be seen in the fact that they equivalently characterize the solutions of an equilibrium problem. We also prove that for some particular instances of ( G E P C ) the gap functions we introduce here become among others the celebrated Auslender’s and Giannessi’s gap functions.

ε-Conjugate maps and ε-conjugate duality in vector optimization with set-valued maps

The aim of this article is to develop an approximate (ε)-conjugate duality theory for a general vector optimization problem with set-valued maps on the basis of ε-weak efficiency. We first introduce the concepts of ε-conjugate maps and ε-weak subgradients for set-valued maps and establish several properties of them. Then we introduce an ε-conjugate dual problem for the vector set-valued optimization problem. Finally, we establish a weak duality theorem and a strong duality theorem for the relationship between the primal and the dual problems.

Notes on free lunch in the limit and pricing by conjugate duality theory

King and Korf [KingKorf01] introduced, in the framework of a discrete- time dynamic market model on a general probability space, a new concept of arbitrage called free lunch in the limit which is slightly weaker than the common free lunch. The definition was motivated by the attempt at proposing the pricing theory based on the theory of conjugate duality in optimization. We show that this concept of arbitrage fails to have a basic property of other common concepts used in pricing theory – it depends on the underlying probability measure more than through its null sets. However, we show that the interesting pricing results obtained by conjugate duality are still valid if it is only assumed that the market admits no free lunch rather than no free lunch in the limit.

An alternative formulation for a new closed cone constraint qualification

We give an alternative formulation for the so-called closed cone constraint qualification ( CCCQ) related to a convex optimization problem in Banach spaces recently introduced in the literature. This new formulation allows to prove in a simple way that ( CCCQ) is weaker than some generalized interior-point constraint qualifications given in the past. By means of some insights from the theory of conjugate duality we also show that strong duality still holds under some weaker hypotheses than the ones considered so far in the literature.

Optimization with set relations: conjugate duality

The aim of this article is to develop a conjugate duality theory, based on set relation approach, for convex set-valued maps. The basic idea is to understand a convex set-valued map as a function with values in the space of closed convex subsets of . The usual inclusion of sets provides a natural ordering relation in this space. Infimum and supremum with respect to this ordering relation can be expressed with the aid of union and intersection. Our main result is a strong duality assertion formulated along the lines of classical duality theorems for extended real-valued convex functions.

A Bilevel Programming Method for Pipe Network Optimization

This paper is concerned with the optimal dimensioning problem in which the layoutof a pipe network is given but several available diameters can be selected for each pipe. The purpose is to minimize total cost while satisfying certain restrictions. A new method is proposed, the main features of which are to treat the problem as a bilevel programming problem, then to simplify the lower level problem by conjugate duality theory and to handle the upper level problem by its piecewise linear and convex nature. Some numerical testing results are also presented.

Eigenvalue optimization

Optimization problems involving eigenvalues arise in many different mathematical disciplines. This article is divided into two parts. Part I gives a historical account of the development of the field. We discuss various applications that have been especially influential, from structural analysis to combinatorial optimization, and we survey algorithmic developments, including the recent advance of interior-point methods for a specific problem class: semidefinite programming. In Part II we primarily address optimization of convex functions of eigenvalues of symmetric matrices subject to linear constraints. We derive a fairly complete mathematical theory, some of it classical and some of it new. Using the elegant language of conjugate duality theory, we highlight the parallels between the analysis of invariant matrix norms and weakly invariant convex matrix functions. We then restrict our attention further to linear and semidefinite programming, emphasizing the parallel duality theory and comparing primal-dual interior-point methods for the two problem classes. The final section presents some apparently new variational results about eigenvalues of nonsymmetric matrices, unifying known characterizations of the spectral abscissa (related to Lyapunov theory) and the spectral radius (as an infimum of matrix norms).

Inference in infinite‐dimensional inverse problems: Discretization and duality

Many techniques for solving inverse problems involve approximating the unknown model, a function, by a finite‐dimensional “discretization” or parametric representation. The uncertainty in the computed solution is sometimes taken to be the uncertainty within the parametrization; this can result in unwarranted confidence. The theory of conjugate duality can overcome the limitations of discretization within the “strict bounds” formalism, a technique for constructing confidence intervals for functionals of the unknown model incorporating certain types of prior information. The usual computational approach to strict bounds approximates the “primal” problem in a way that the the resulting confidence intervals are at most long enough to have the nominal coverage probability. There is another approach based on “dual” optimization problems that gives confidence intervals with at least the nominal coverage probability. The pair of intervals derived by the two approaches bracket a correct confidence interval. The theory is illustrated with gravimetric, seismic, geomagnetic, and helioseismic problems and a numerical example in seismology.

Conjugate duality in vector optimization

In this paper we develop a conjugate duality theory in vector optimiza- tion. Conjugate duality was fully developed in scalar optimization by Rockafellar [3 ] and provides a unified framework for the duality theory. The author and Sawaragi extended it to the case of multiobjective optimization by introducing some new concepts such as conjugate maps and subgradients for vector-valued, set-valued maps [4,6]. Their results are based on the efficiency (Pareto maximality). Kawasaki refined their results and obtained a reciprocal duality by introducing the concept of “supremum set” in an extended Euclidean space [ 1,2]. His supremum is based on the weak efficiency (weak Pareto maximality) of the closure of a set in an extended sense. The author [S] newly defined the supremum of a set in the extended multi-dimensional Euclidean space on the basis of weak ehiciency through several discussions on supremum. The definition satisfies some desirable fundamental properties. It can be extended to the case of general partially ordered linear topological spaces in a straightforward manner. Based on this definition of supremum, some useful concepts such as conjugate maps and subgradients are introduced for vector-valued, set-valued maps in this paper. These concepts enable us to develop the conjugate duality in vector optimization. Although the results obtained are quite similar to the earlier works by the author and Sawaragi [4] or Kawasaki [2], our new approach makes the proofs much easier and more understandable, and provides a better version of conjugate duality.

A Unified Approach to Term Structure Estimation: A Methodology for Estimating the Term Structure in a Market with Frictions

This paper develops a methodology for term structure estimation from a no-arbitrage condition in markets with frictions. The methodology unifies existing estimation procedures, such as the regression and linear programming approaches, and substantially broadens the class of useful estimation techniques. The estimators derived in this way are capable of reflecting actual market conditions, such as the asymmetry in the tax treatment of long and short positions and the higher financial cost of establishing a short position. The methodology is derived by applying the conjugate duality theory of mathematical programming.

Stop rule and supremum expectations of i.i.d. random variables: A complete comparison by conjugate duality

A complete comparison is made between the value V( X 1,…, X n ) = sup{ EX t : t is a stop rule for X 1,…, X n } and E(max j ≤ n X j ) for all uniformly bounded sequences of i.i.d. random variables X 1, …, X n . Specifically, the set of ordered pairs {( x, y): x = V( X 1, …, X n ) and y = E(max j≤ n X j ) for some i.i.d.r.v.'s X 1,…, X n taking values in [0, 1]} is precisely the set {( x, y): x≤ y≤ Γ n ( x); 0 ≤ x≤1}, where the upper boundary function Γ n is given in terms of recursively defined functions. The result yields families of inequalities for the “prophet” problem, relating the motal's value of a game V( X 1, …, X n ) to the prophet's value of the game E(max j≤ n X j ). The proofs utilize conjugate duality theory, probabilistic convexity arguments, and functional equation analysis. Asymptotic analysis of the “prophet” regions and inequalities is also given.