General Duality Theory Research Articles

Duality and Free Measures in Vector Spaces, the Spectral Theory of Actions of Non-Locally Compact Groups

The paper presents a general duality theory for vector measure spaces taking its origin in author’s papers written in the 1960s. The main result establishes a direct correspondence between the geometry of a measure in a vector space and properties of the space of measurable linear functionals on this space regarded as closed subspaces of an abstract space of measurable functions. An example of useful new features of this theory is the notion of a free measure and its applications.

On duality of algebraic quantum groupoids

Like quantum groups, quantum groupoids frequently appear in pairs of mutually dual objects. We develop a general Pontrjagin duality theory for quantum groupoids in the algebraic setting that extends Van Daele's duality theory for multiplier Hopf algebras and overcomes the finiteness restrictions of the approach of Kadison, Szlachányi, Böhm and Schauenburg. Our construction is based on the integration theory for multiplier Hopf algebroids and yields, as a corollary, a duality theory for weak multiplier Hopf algebras with integrals. We compute the duals in several examples and introduce morphisms of multiplier Hopf algebroids to succinctly describe their structure. Moreover, we show that such morphisms preserve the antipode.

Duality theory for portfolio optimisation under transaction costs

We consider the problem of portfolio optimisation with general c adl ag price processes in the presence of proportional transaction costs. In this context, we develop a general duality theory. In particular, we prove the existence of a dual optimiser as well as a shadow price process in an appropriate generalised sense. This shadow price is dened by means of a \sandwiched process consisting of a predictable and an optional strong supermartingale, and pertains to all strategies that remain solvent under transaction costs. We provide examples showing that, in the general setting we study, the shadow price processes have to be of such a generalised form. MSC 2010 Subject Classication: 91G10, 93E20, 60G48

Toward a neoclassical theory of sustainable consumption: Eight golden age propositions

Popular trends in ecological economics increasingly consign neoclassical economics to the sidelines of modern-day relevancy. The neoclassical tradition is often seen as reliant for its authenticity on a presumption of human avarice – both unbridled consumerism and corporate cupidity – and demanding for its real-world applicability an assumption of continuous economic growth in a world of hard limits.This article examines the question of whether neoclassical theory could instead provide keys to deeper understanding of sustainable consumption. By combining in a single framework neoclassical growth theory, general equilibrium theory and duality theory – and by explicitly considering leisure time – the analysis demonstrates that neoclassical economics yields several useful insights bearing on long-term sustainability. The analysis confirms several tenets of ecological economics and challenges others.Eight propositions emerge from this analysis that could help speed the development of a robust neoclassical theory of sustainable consumption, here branded “golden age” propositions as they strongly echo the “Golden Rule” discoveries of Edmund Phelps.

A GENERAL DUALITY THEORY FOR CLONES

Inspired by work of Mašulović, we outline a general duality theory for clones that will allow us to dualize any given clone, together with its relational counterpart and the relationship between them. Afterwards, we put the approach to work and illustrate it by producing some specific results for concrete examples as well as some general results that come from studying the duals of clones in a rather abstract fashion.

A unified approach to multiple stopping and duality

The main approaches to dual representations of multiple stopping problems are the marginal and pure martingale approaches of Meinshausen and Hambly (2004) [17] and Schoenmakers (2010) [20], respectively. We show that these dual representations, as well as their more recent extensions to problems with volume constraints, can be derived in a simple unified manner using the recently developed general duality theory based on information relaxations. We also derive pure martingale representations for other multiple stopping problems, including problems with refractive index constraints.

Approaches to Conditional Risk

We present and compare two different approaches to conditional risk measures. One approach draws from convex analysis in vector spaces and presents risk measures as functions on Lp spaces, while the other approach utilizes module-based convex analysis where conditional risk measures are defined on Lp type modules. Both approaches utilize general duality theory for vector valued convex functions in contrast to the current literature in which we find ad hoc dual representations. By presenting several applications such as monotone and (sub)cash invariant hulls with corresponding examples we illustrate that module-based convex analysis is well suited to the concept of conditional risk measures.

A Unified Approach to Multiple Stopping and Duality

The main approaches to dual representations of multiple optimal stopping problems are the marginal and pure martingale approaches of Meinhausen and Hambly (2004) and Schoenmakers (2010), respectively. In this paper we show that these dual representations can be derived in a simple unified manner using the general duality theory based on information relaxations that was developed independently by Brown, Smith and Sun (2010) and Rogers (2007). We also show that recent extensions of the marginal dual representation to problems with volume constraints are easily handled using this unified approach. We also derive pure martingale representations for the multiple stopping problem with volume constraints and the multiple stopping problem with so-called refractive index constraints. These latter two representations are new and they can be used in practice for the pricing of swing options in electricity markets.

The algebra of set functions II: An enumerative analogue of Hall’s theorem for bipartite graphs

Triesch (1997) [25] conjectured that Hall’s classical theorem on matchings in bipartite graphs is a special case of a phenomenon of monotonicity for the number of matchings in such graphs. We prove this conjecture for all graphs with sufficiently many edges by deriving an explicit monotonic formula counting matchings in bipartite graphs. This formula follows from a general duality theory which we develop for counting matchings. Moreover, we make use of generating functions for set functions as introduced by Lass [20], and we show how they are useful for counting matchings in bipartite graphs in many different ways.

The algebra of set functions I: The product theorem and duality

We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.

Duality for max-separable problems

In this paper we propose a general duality theory for a class of so called ‘max-separable’ optimization problems. In such problems functions h:Rk → R of the form h(x1, . . . , xk) = maxj hj(xj), occur both as objective functions and as constraint functions (hj are assumed to be strictly increasing functions of one variable). As a result we obtain pairs of max-separable optimization problems, which possess both weak and strong duality property without a duality gap.

Lipschitz continuity and duality for dynamic oligopolistic market equilibrium problem with memory term

We are concerned with an infinite dimensional variational inequality which is connected with the dynamic oligopolistic market equilibrium problem. We will provide existence theorems and show, under minimal assumptions on the data, the Lipschitz continuity of the solution. Moreover a general duality theory is provided overcoming the difficulty of the voidness of the interior of the ordering cone which defines the cone constraints.

Approaches to Conditional Risk

We present and compare two different approaches to conditional risk measures. One approach draws from convex analysis in vector spaces and presents risk measures as functions on Lp spaces, while the other approach utilizes module-based convex analysis where conditional risk measures are defined on Lp type modules. Both approaches utilize general duality theory for vector valued convex functions in contrast to the current literature in which we find ad hoc dual representations. By presenting several applications such as monotone and (sub)cash invariant hulls with corresponding examples we illustrate that module-based convex analysis is well suited to the concept of conditional risk measures.

Analysis of SITA policies

We analyze the performance of Size Interval Task Assignment (SITA) policies, for multi-host assignment in a non-preemptive environment. Assuming Poisson arrivals, we provide general bounds on the average waiting time independent of the job size distribution. We establish a general duality theory for the performance of SITA policies. We provide a detailed analysis of the performance of SITA systems when the job size distribution is Bounded Pareto and the range of job sizes tends to infinity. In particular, we determine asymptotically optimal cutoff values and provide asymptotic formulas for average waiting time and slowdown. We compare the results with the Least Work Remaining policy and compute which policy is asymptotically better for any given set of parameters. In the case of inhomogeneous hosts, we determine their optimal ordering.

Analysis of size interval task assignment policies

We analyze the performance of Size Interval task assignment (SITA) scheduling policies, for multi-host scheduling in a non-preemptive environment. We establish a general duality theory for the performance analysis of SITA policies. When the job size distribution is Bounded Pareto and the range of job sizes tends to infinity. we determine asymptotically optimal cutoff values and provide asymptotic formulas for average waiting time and slowdown. In the case of inhomogeneous hosts we determine their optimal ordering. We also consider TAGS policies. We provide a general formula that describes their load handling capabilities and examine their performance when the job size distribution is Bounded Pareto.

On the theory of Lagrangian duality

It is shown that a general Lagrangian duality theory for constrained extremum problems can be drawn from a separation scheme in the Image Space, namely in the space where the functions of the given problem run.

Robust utility maximization in a stochastic factor model

SUMMARY We give an explicit PDE characterization for the solution of a robust utility maximization problem in an incomplete market model, whose volatility, interest rate process, and long-term trend are driven by an external stochastic factor process. The robust utility functional is defined in terms of a HARA utility function with negative risk aversion and a dynamically consistent coherent risk measure, which allows for model uncertainty in the distributions of both the asset price dynamics and the factor process. Our method combines two recent advances in the theory of optimal investments: the general duality theory for robust utility maximization and the stochastic control approach to the dual problem of determining optimal martingale measures.

Canonical Dual Transformation Method and Generalized Triality Theory in Nonsmooth Global Optimization

This paper presents, within a unified framework, a potentially powerful canonical dual transformation method and associated generalized duality theory in nonsmooth global optimization. It is shown that by the use of this method, many nonsmooth/nonconvex constrained primal problems in \realn can be reformulated into certain smooth/convex unconstrained dual problems in \realm with m≤slant n and without duality gap, and some NP-hard concave minimization problems can be transformed into unconstrained convex minimization dual problems. The extended Lagrange duality principles proposed recently in finite deformation theory are generalized suitable for solving a large class of nonconvex and nonsmooth problems. The very interesting generalized triality theory can be used to establish nice theoretical results and to develop efficient alternative algorithms for robust computations.

Finite deformation beam models and triality theory in dynamical post-buckling analysis

Two new finitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola–Kirchhoff’s type stress only) in finite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Duffing equations are obtained.

Palm Measure Duality and Conditioning in Regenerative Sets

For a simple point process $\Xi$ on a suitable topological space, the associated Palm distribution at a point s may be approximated by the conditional distribution, given that $\Xi$ hits a small neighborhood of $s$. To study the corresponding approximation problem for more general random sets, we develop a general duality theory, which allows the Palm distributions with respect to an associated random measure to be expressed in terms of conditional densities with suitable martingale and continuity properties. The stated approximation property then becomes equivalent to a certain asymptotic relation involving conditional hitting probabilities. As an application, we consider the Palm distributions of regenerative sets with respect to their local time random measures.