Posynomial Geometric Programming Problems with Multiple Parameters

Geometric programming problem is a powerful tool for solving some special type non-linear programming problems. It has a wide range of applications in optimization and engineering for solving some complex optimization problems. Many applications of geometric programming are on engineering design problems where parameters are estimated using geometric programming. When the parameters in the problems are imprecise, the calculated objective value should be imprecise as well. In this paper we have developed a method to solve geometric programming problems where the exponent of the variables in the objective function, cost coefficients and right hand side are multiple parameters. The equivalent mathematical programming problems are formulated to find their corresponding value of the objective function based on the duality theorem. By applying a variable separable technique the multi-choice mathematical programming problem is transformed into multiple one level geometric programming problem which produces multiple objective values that helps engineers to handle more realistic engineering design problems

Geometric duality and parametric duality for multiple objective linear programs are equivalent

In 2011, Luc introduced parametric duality for multiple objective linear programs. He showed that geometric duality, introduced in 2008 by Heyde and L\"ohne, is a consequence of parametric duality. We show the converse statement: parametric duality can be derived from geometric duality. We point out that an easy geometric transformation embodies the relationship between both duality theories. The advantages of each theory are discussed.Comment: 10 pages, 2 figures, new version is just a minor revisio

The vector linear program solver Bensolve -- notes on theoretical background

Bensolve is an open source implementation of Benson's algorithm and its dual variant. Both algorithms compute primal and dual solutions of vector linear programs (VLP), which include the subclass of multiple objective linear programs (MOLP). The recent version of Bensolve can treat arbitrary vector linear programs whose upper image does not contain lines. This article surveys the theoretical background of the implementation. In particular, the role of VLP duality for the implementation is pointed out. Some numerical examples are provided.Comment: 17 pages, 5 tables, 1 figur

Multi-objective Geometric Programming Problem With Weighted Mean Method

Geometric programming is an important class of optimization problems that enable practitioners to model a large variety of real-world applications, mostly in the field of engineering design. In many real life optimization problem multi-objective programming plays a vital role in socio-economical and industrial optimizing problems. In this paper we have discussed the basic concepts and principle of multiple objective optimization problems and developed geometric programming (GP) technique to solve this optimization problem using weighted method to obtain the non-inferior solutions.Comment: Pages IEEE format, International Journal of Computer Science and Information Security, IJCSIS February 2010, ISSN 1947 5500, http://sites.google.com/site/ijcsis

Pareto optimality conditions and duality for vector quadratic fractional optimization problems

One of the most important optimality conditions to aid to solve a vector optimization problem is the first-order necessary optimality condition that generalizes the Karush-Kuhn-Tucker condition. However, to obtain the sufficient optimality conditions, it is necessary to impose additional assumptions on the objective functions and in the constraint set. The present work is concerned with the constrained vector quadratic fractional optimization problem. It shows that sufficient Pareto optimality conditions and the main duality theorems can be established without the assumption of generalized convexity in the objective functions, by considering some assumptions on a linear combination of Hessian matrices instead. The main aspect of this contribution is the development of Pareto optimality conditions based on a similar second-order sufficient condition for problems with convex constraints, without convexity assumptions on the objective functions. These conditions might be useful to determine termination criteria in the development of algorithms.Comment: 22 page

On Unified Modeling, Canonical Duality-Triality Theory, Challenges and Breakthrough in Optimization

A unified model is addressed for general optimization problems in multi-scale complex systems. Based on necessary conditions and basic principles in physics, the canonical duality-triality theory is presented in a precise way to include traditional duality theories and popular methods as special applications. Two conjectures on NP-hardness are discussed, which should play important roles for correctly understanding and efficiently solving challenging real-world problems. Applications are illustrated for both nonconvex continuous optimization and mixed integer nonlinear programming. Misunderstandings and confusion on some basic concepts, such as objectivity, nonlinearity, Lagrangian, and Lagrange multiplier method are discussed and classified. Breakthrough from recent false challenges by C. Z\u{a}linescu and his co-workers are addressed. This paper will bridge a significant gap between optimization and multi-disciplinary fields of applied math and computational sciences.Comment: 28 pages, 2 figure

A vector linear programming approach for certain global optimization problems

Global optimization problems with a quasi-concave objective function and linear constraints are studied. We point out that various other classes of global optimization problems can be expressed in this way. We present two algorithms, which can be seen as slight modifications of Benson-type algorithms for multiple objective linear programs (MOLP). The modification of the MOLP algorithms results in a more efficient treatment of the studied optimization problems. This paper generalizes results of Schulz and Mittal on quasi-concave problems and Shao and Ehrgott on multiplicative linear programs. Furthermore, it improves results of L\"ohne and Wagner on minimizing the difference $f=g-h$ of two convex functions $g$, $h$ where either $g$ or $h$ is polyhedral. Numerical examples are given and the results are compared with the global optimization software BARON.Comment: same content like journal version; difference to previous version: some typos in the text correcte

Canonical Duality-Triality Theory: Bridge Between Nonconvex Analysis/Mechanics and Global Optimization in Complex Systems

Canonical duality-triality is a breakthrough methodological theory, which can be used not only for modeling complex systems within a unified framework, but also for solving a wide class of challenging problems from real-world applications. This paper presents a brief review on this theory, its philosophical origin, physics foundation, and mathematical statements in both finite and infinite dimensional spaces, with emphasizing on its role for bridging the gap between nonconvex analysis/mechanics and global optimization. Special attentions are paid on unified understanding the fundamental difficulties in large deformation mechanics, bifurcation/chaos in nonlinear science, and the NP-hard problems in global optimization, as well as the theorems, methods, and algorithms for solving these challenging problems. Misunderstandings and confusions on some basic concepts, such as objectivity, nonlinearity, Lagrangian, and generalized convexities are discussed and classified. Breakthrough from recent challenges and conceptual mistakes by M. Voisei, C. Zalinescu and his co-worker are addressed. Some open problems and future works in global optimization and nonconvex mechanics are proposed.Comment: 43 pages, 4 figures. appears in Mathematics and Mechanics of Solids, 201

Lexicographic Multi-objective Geometric Programming Problems

A Geometric programming (GP) is a type of mathematical problem characterized by objective and constraint functions that have a special form. Many methods have been developed to solve large scale engineering design GP problems. In this paper GP technique has been used to solve multi-objective GP problem as a vector optimization problem. The duality theory for lexicographic geometric programming has been developed to solve the problems with posynomial in objectives and constraints.Comment: International Journal of Computer Science Issues, IJCSI Volume 6, Issue 2, pp20-24, November 200

On Linear Programming for Constrained and Unconstrained Average-Cost Markov Decision Processes with Countable Action Spaces and Strictly Unbounded Costs

We consider the linear programming approach for constrained and unconstrained Markov decision processes (MDPs) under the long-run average cost criterion, where the class of MDPs in our study have Borel state spaces and discrete countable action spaces. Under a strict unboundedness condition on the one-stage costs and a recently introduced majorization condition on the state transition stochastic kernel, we study infinite-dimensional linear programs for the average-cost MDPs and prove the absence of duality gap and other optimality results. Our results do not require a lower-semicontinuous MDP model and as such, they can be applied to countable action space MDPs where the dynamics and one-stage costs are discontinuous in the state variable. The proofs of these results make use of the continuity property of Borel measurable functions asserted by Lusin's theorem.Comment: 33 pages; some sections reordered and minorly revised; submitte