Linear Mathematical Programming

This paper explores the most urgent task to initialize a virtual network in the network virtualization environment, which always called the virtual network embedding. Different to the previous works to take the embedding process as two steps: node mapping stage and then edge mapping stage, this paper contribute to solve these processes all in one discrete nonlinear combined optimization. Based on modeling the virtual network (VN) request and substrate network (SN), this paper proposed mathematical description to the embedding process which building the relationship between node mapping and edge mapping stages. After that, a virtual network embedding cost is introduced as an objective function, and then the virtual network embedding can be concluding to minimize the objective cost as a mathematical programming problem. The obtained programming results in LINGO9.0 show that the proposed method can finish a virtual network embedding by mathematical programming in one step.

The class of combinatorial optimization problems over polyhedral-spherical sets is considered. The results of theory of convex extensions are generalized to certain classes of functions defined on sphere-located and vertex-located sets. The original problem is equivalently formulated as a mathematical programming problem with convex both objective function and functional constraints. A numerical illustration and possible applications of the results to solving combinatorial optimization problems are given.

The necessary and sufficient conditions for single extremality in non-convex problems of mathematical programming

Chapter 1 - Introduction

Mathematical programming procedures to optimize the collector field subsystem of a power tower system

On the Use of Lagrange Duality in Multilevel Optimal Control

An η-approximation approach to duality in mathematical programming problems involving r-invex functions

Aim/purpose – Shelf space is one of the most important tools for attracting customers’ attention in a retail store. This paper aims to develop a practical shelf space allocation model with visible vertical and horizontal categories. and formulate it in linear and non- -linear forms. Design/methodology/approach – The research is mainly based on operational research. Simulation, mathematical optimization, and linear and nonlinear programming methods are mainly used. Special attention is given to the decision variables and constraints. Changing the dimensioning of the decision variables results in an improvement in the formulation of the problem, which in turn allows for obtaining an optimal solution. Findings – A comparison of the developed shelf space allocation models with visible vertical and horizontal categories in linear and nonlinear forms is presented. The compu- tational experiments were performed with the help of CPLEX solver, which shows that the optimal solution of the linear problem formulation was obtained within a couple of seconds. However, a nonlinear form of this problem found the optimal solution only in 19 out of 45 instances. An increase in the time limits slightly improves the performance of the solutions of the nonlinear form. Research implications/limitations – The main implication of research results for sci- ence is related to the possibility of determining an optimal solution to the initially formu- lated nonlinear shelf space allocation problem. The main implication for practice is to take into consideration the practical constraints based on customers’ requirements. The main limitations are the lack of storage conditions and holding time constraints. Originality/value/contribution – The main contribution is related to developing math- ematical models that consider simultaneous categorization of products vertically, based on one characteristic, and horizontally, based on another characteristic. Contribution is also related to extending the shelf space allocation theory with the shelf space allocation problem model in relation to four sets of constraints: shelf constraints, product con- straints, orientation constraints, and band constraints. Keywords: Retailing, decision making/process, merchandising, shelf space allocation, planogram. JEL Classification: C61, L81.

This paper establishes a direct method for solving variational problems via a set of Radial basis functions (RBFs) with Gauss-Chebyshev collocation centers. The method consist of reducing a variational problem into a mathematical programming problem. The authors use some optimization techniques to solve the reduced problem. Accuracy and stability of the multiquadric, Gaussian and inverse multiquadric RBF is examined and compared by some numerical experiments.

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

We still face situations where there are different goals and constraints that need to be optimized in mathematical programming problems, but their value cannot be easily determined. However, when uncertainty is included in the model, the translation of real-world problems into mathematical models becomes more difficult. Given the nature of the multi-objective conditions under hybrid uncertainty, it is important to adapt to the model. In some research studies, only purpose functions or constraints are taken into account vaguely. But in this research study, objectives, constraints, and coefficients were calculated based on fuzzy random data. Goal and constraint functions should be constructed by analyzing past data, then determining constraint functions based on volatile hybrid environments. In the building constraint function, “about equal to” should be converted into fuzzy mathematical symbols and figures to express. The final step is provided to find the best answer using scalable methods and maximum operators to complete multi-criteria goal programming. The algorithm uses the concept of satisfaction to multi-objective optimization. The problem model was also developed by means of a fuzzy random regression approach. From that, we emphasize that the proposed method has significant advantages in solving multi-objective problems in which fuzzy random information coexisting.

My research in optimization at the Department of Systems and Computer Engineering at Carleton University has brought me into contact with researchers in the Department of Electronics, our sister department in the general field of Electrical and Computer Engineering (ECE), which has a major research thrust in computer-aided design (CAD) of very large scale integrated (VLSI) circuits. I have come to realize that the design of VLSI systems consists in large part of solving incredibly massive mixed-integer nonlinear optimization problems, together with enormous circuit simulations. This is impossible without the use of CAD techniques. Over time, ECE-CAD researchers have both borrowed useful standard techniques from operations research (OR) and have invented their own, often to deal with the sheer scale and complexity of the design problems they face. Interaction between the OR and ECE-CAD communities would seem to be a natural development. Coincidentally, the INFORMS Journal on Computing had an area entitled “High Performance Computation” that covered (i) the solution of OR problems using new computing technologies, (ii) the application of OR techniques in the design and use of highperformance computing and communication systems, and (iii) solution methods for ultra-large-scale OR applications. The viability of this area was debated during 2003 as some of its aspects migrated to other areas (e.g., the new “Telecommunications and Electronic Commerce” area), and the number of submissions declined. The area was eventually closed, although I argued that the overlap between OR and ECE-CAD, essentially covering items (ii) and (iii), was a very active research area. This Special Cluster of papers was conceived as a way to test that argument. I was recruited at the same time to prepare a tutorial on ECE-CAD for the 2004 INFORMS Annual Meeting (see John W. Chinneck, Michel Nakhla, and Q. J. Zhang 2004. Computer-aided design for electrical and computer engineering. H. J. Greenberg, ed. Tutorials on Emerging Methodologies and Applications in Operations Research. Springer, New York, 6-1 to 6-44). The preparation of the article reinforced my observation about the research overlap between OR and ECE-CAD. A search of the electrical engineering literature for 2000 through early 2004 turned up 46,725 papers mentioning “simulate” or “simulation” as keywords in the abstract, 14,216 papers mentioning “optimization” or “optimize,” and relatively smaller numbers for specific techniques such as “neural network” (6,251), “genetic algorithm” (2,603), “linear programming” (576), “simulated annealing” (448), and “branch and bound” (199). One surprise was the relatively small number of papers using the generic keywords “mathematical programming” (68) or “operations research” (37). The general conclusion of the tutorial article is that the OR and the ECE-CAD communities have much to offer each other. This is certainly the case for the papers gathered in this Special Cluster. We see known OR techniques adapted for use in ECECAD. In “Integer Linear Programming Models for Global Routing,” Behjat et al. apply integer linear programming in a heuristic to solve enormous NPhard connection routing problems for VLSI circuits. In “Task Scheduling in a Finite-Resource, Reconfigurable Hardware/Software Codesign Environment,” Loo and Wells use simulated annealing, genetic algorithms, and random search techniques to solve scheduling problems in hardware-software co-design. We also see new techniques specifically developed by the ECE-CAD community to deal with problems of extreme scale. In “A Projection-Based Reduction Approach to Computing Sensitivity of Steady-State Response of Nonlinear Circuits,” Pai et al. develop methods for sensitivity analysis in extremely large nonlinear programs, especially those in which the objective function is very costly to evaluate. The methods have special relevance for simulation-based

We give an alternative proof of the optimality of the solution for the deterministic EPQ with partial backordering (EPQ-PBO) [Omega, vol. 37, no. 3, pp. 624–636, 2009]. Our proof is based on the mathematical programming theory. We also demonstrate the determination of the optimal decision policy through solving the corresponding mathematical programming problem. We indicate that the same approach can be used within other inventory models with partial backordering, and we consider additional models.

Abstract. In this paper, some omissions in Mishra and Lai [13], havebeen pointed out and their corrective measures have been discussed briefly. 1. IntroductionA pair of primal and dual problems in mathematical programming is calledsymmetric if the dual of the dual is the primal problem. Dorn [6] introducedthe concept of symmetric duality in quadratic programming. His results wereextended to nonlinear convex programming problems by Dantzig et al. [4] andlater by Bazaraa and Goode [3] over arbitrary cones.Mangasarian [11] introduced the concept of second-order duality for non-linear problems. Since then, many authors [1, 2, 7, 9, 15, 16] have workedon second-order symmetric duality. Mishra and Lai [13] studied Mond-Weirtype second-order multiobjective symmetric duality for the following pair ofproblems:(P) K−minimize f(x,y)−12p T ∇ yy f(x,y)psubject to −∇ y (λ T f)(x,y) −∇ yy (λ T f)(x,y) ∈ C ∗2 ,y T [∇ y (λ T f)(x,y) +∇ yy (λ T f)(x,y)] > 0,λ ∈ K ∗ , x ∈ C 1 .(D) K−maximize f(u,v)−12q

Necessary conditions of optimality are obtained for general mathematical programming problems on a product space. The cost functional is locally Lipschitz and the constraints are expressed as inclusion relations with unbounded linear operators and multivalued term. The abstract result is applied to an optimal control problem governed by an elliptic differential inclusion.