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Primal Problem (primal + problem)

Distribution by Scientific Domains
Business, Economics, Finance and Accounting40%

Selected Abstracts

A dual extremum principle in thermodynamics ,

AICHE JOURNAL, Issue 8 2007
Alexander Mitsos
Abstract Phase equilibria of multicomponent mixtures are considered and a reinterpretation of the Gibbs tangent plane stability criterion is proposed via Lagrangian duality. The starting point is the natural primal problem of minimizing the Gibbs free energy subject to material balance. The stable phase split is the solution of the corresponding dual problem, providing a necessary and sufficient dual extremum principle. Only in the absence of duality gap is the physical phase split also the solution of the primal problem. The only requirements are continuity of the Gibbs free energy and the trivial requirement that each species is present in the overall composition. The number of phases is permitted to be infinite, and does not need to be known a priori. No assumption is made on the presence of all species in all phases. Case studies are presented based on the NRTL and UNIQUAC activity coefficient model. © 2007 American Institute of Chemical Engineers AIChE J, 2007 [source]

Lagrange Multipliers as Marginal Rates of Substitution in Multi-Constraint Optimization Problems

METROECONOMICA, Issue 1 2001
Christian E. Weber
This paper shows that, when a function is optimized subject to several binding constraints, some of the Lagrange multipliers in the dual problems can be interpreted as marginal rates of substitution among certain arguments in the generalized indirect objective function for the primal problem. It also shows how to calculate these Lagrange multipliers from observable price,quantity data. Three particular examples are discussed: a firm that minimizes costs subject to both fixed output and rationing constraints, a household that maximizes utility subject to both income and time constraints, and portfolio choice under uncertainty treated as a multiple constraint optimization problem. [source]

Optimal carbon source switching strategy for the production of PHA copolymers

AICHE JOURNAL, Issue 3 2001
Nikolaos V. Mantzaris
During polymerization in a nongrowing cell population of Ralstonia eutropha, alternating between two different carbon sources (fructose and fructose/valeric acid) could lead to the production of block copolymers consisting of blocks of homo-poly-3-hydroxybutyrate (PHB) and polyhydroxybutyrate-co-valerate (PHBV) copolymer. The problem of finding the optimal number of carbon source switches and corresponding switching times that maximize the final concentration of diblock copolymers (PHB-PHBV and PHBV-PHB) was addressed. It was mathematically formulated in the mixed-integer nonlinear programming (MINLP) framework, which allows the decomposition of the original problem into the primal and master problems. The primal problem corresponds to the original problem for a fixed number of carbon source switches, whereas the master problem consists of finding the number of carbon source switches that maximizes the optimum solutions of all possible primal problems. The global optimum was obtained for 39 carbon source switches. It corresponds to a mass fraction of 50.6% of final diblock copolymer concentration over the final total polymer concentration. [source]

DUALITY IN OPTIMAL INVESTMENT AND CONSUMPTION PROBLEMS WITH MARKET FRICTIONS

MATHEMATICAL FINANCE, Issue 2 2007
I. Klein
In the style of Rogers (2001), we give a unified method for finding the dual problem in a given model by stating the problem as an unconstrained Lagrangian problem. In a theoretical part we prove our main theorem, Theorem 3.1, which shows that under a number of conditions the value of the dual and primal problems is equal. The theoretical setting is sufficiently general to be applied to a large number of examples including models with transaction costs, such as Cvitanic and Karatzas (1996) (which could not be covered by the setting in Rogers [2001]). To apply the general result one has to verify the assumptions of Theorem 3.1 for each concrete example. We show how the method applies for two examples, first Cuoco and Liu (1992) and second Cvitanic and Karatzas (1996). [source]