We consider the problem of optimal consumption of multiple goods in incomplete semimartingale markets. We formulate the dual problem and identify conditions that allow for the existence and uniqueness of the solution, and provide a characterization of the optimal consumption strategy in terms of the dual optimizer. We illustrate our results with examples in both complete and incomplete models. In particular, we construct closed-form solutions in some incomplete models.

We consider the problem of optimal investment with intermediate consumption in a general semimartingale model of an incomplete market, with preferences being represented by a utility stochastic field. We show that the key conclusions of the utility maximization theory hold under the assumptions of no unbounded profit with bounded risk and of the finiteness of both primal and dual value functions.

This article proposes that duality theory plays a role in obtaining more nuanced and textured insights into the complex, paradoxical stability–change nexus by illustrating how tensions are managed not through definitive resolution toward one pole or the other, but through improvised boundary heuristics that establish a broad conforming imperative while opening up enabling mechanisms. Duality thinking also reinforces the need to discard assumptions about opposing values, instead replacing them with an appreciation of complementary concepts. The article explores the characteristics of dualities to allow managers to chart what they are seeking from their management interventions and subsequent choices in structural support systems. A key benefit of identifying and explaining duality characteristics comes in attempting to understand how to mediate between two contradictory dimensions of organizing, such as continuity and change. Our argument is that both need to be encouraged, but this requires a particular mindset where the problem of mediation viewed as the need to work towards simultaneity and synergistic mutuality rather than resolution of action between the two opposing dimensions.

A variational principle is established to provide a new formulation for convex Hamiltonian systems. Using this formulation, we obtain some existence results for second-order Hamiltonian systems with a variety of boundary conditions, including nonlinear ones.

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at -∞, we prove that the utility-based superreplication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite loss-entropy. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is a proof of the duality between the cone of utility-based superreplicable contingent claims and the cone generated by pricing measures with finite loss-entropy.

This paper provides KKT and saddle point optimality conditions, dualitytheorems and stability theorems for consistent convex optimization problemsposed in locally convex topological vector spaces. The feasible sets ofthese optimization problems are formed by those elements of a given closedconvex set which satisfy a (possibly infinite) convex system. Moreover, allthe involved functions are assumed to be convex, lower semicontinuous andproper (but not necessarily real-valued). The key result in the paper is thecharacterization of those reverse-convex inequalities which are consequenceof the constraints system. As a byproduct of this new versions of Farkas'lemma we also characterize the containment of convex sets in reverse-convexsets. The main results in the paper are obtained under a suitableFarkas-type constraint qualifications and/or a certain closedness assumption.

This paper examines an intertemporal optimizing consumer or a representative consumer-firm in a deterministic setting subject to a general (either linear or nonlinear) capital accumulation equation. Duality theory is used to recast the Hamilton–Jacobi equation for dynamic optimization in terms of an instantaneous and an intertemporal profit function. An envelope theorem allows derivation of an explicit solution for the value of the costate variable as a function of the state and other variables. The final model form only requires specification of atemporal functions that are linked into a closed-form solution for the optimal dynamic decision variables through a system of contemporaneous simultaneous equations.

We give a revised and updated exposition of the theory of full dualities initiated by Clark, Davey, Krauss and Werner, introducing the (stronger) notion of a strong duality. All known full dualities turn out to be strong. A series of theorems which provide necessary and sufficient conditions for a strong duality to exist is proved. All full dualities in the literature can be obtained from these results and many new strong dualities can be derived. In particular, we show that within congruence distributive varieties every duality can be upgraded to a strong duality. Amongst the new strong dualities are the dualities of Davey, Priestley and Werner for the varieties of pseudocomplemented distributive lattices which are either strong as they stand or can easily be made strong by the addition of partial operations to the dual structures.