A Review of Deterministic Optimization Methods in Engineering and Management

With the increasing reliance on modeling optimization problems in practical applications, a number of theoretical and algorithmic contributions of optimization have been proposed. The approaches developed for treating optimization problems can be classified into deterministic and heuristic. This paper aims to introduce recent advances in deterministic methods for solving signomial programming problems and mixed-integer nonlinear programming problems. A number of important applications in engineering and management are also reviewed to reveal the usefulness of the optimization methods.


Introduction
The field of optimization has grown rapidly during the past few decades.Many new theoretical, algorithmic, and computational contributions of optimization have been proposed to solve various problems in engineering and management.Recent developments of optimization methods can be mainly divided into deterministic and heuristic approaches.Deterministic approaches take advantage of the analytical properties of the problem to generate a sequence of points that converge to a global optimal solution.Heuristic approaches have been found to be more flexible and efficient than deterministic approaches; however, the quality of the obtained solution cannot be guaranteed.Moreover, the probability of finding the global solution decreases when the problem size increases.Deterministic approaches e.g., linear programming, nonlinear programming, and mixed-integer nonlinear programming, etc. can provide general tools for solving optimization problems to obtain a global or an approximately global optimum.With the increasing reliance on Mathematical Problems in Engineering modeling optimization problems in real applications, a number of deterministic methods for optimization problems have been presented.This paper focuses on discussing and reviewing the recent advances in deterministic optimization approaches.
Optimization methods have been applied in different fields, such as finance 1-3 , allocation and location problems 4-6 , engineering design 7-12 , system and database design 13-17 , chemical engineering design and control 18-22 , and molecular biology 23 .For additional literature on real-world applications or developments of optimization methods, readers may refer to the following works.Mockus et al. 24 treated network problems, knapsack, travelling salesman, flow-shop problems, and batch process scheduling problems by Bayesian heuristic approaches.Grossmann 25 discussed the global optimization algorithms and their applications in chemical engineering design.Bomze   Figure 1 gives an overview of the problem types related to optimization problems.Each type of problems has received substantial attention from the practitioners and the researchers in the last few decades.In this paper, we investigate the advances in deterministic global optimization of nonconvex nonlinear programming NLP problems and nonconvex mixed-integer nonlinear programming MINLP problems.For NLP problems, we focus on signomial programming problems that are an important class of NLP problems and have played a crucial role in applications.The rest of this paper is organized as follows.Section 2 discusses the deterministic methods for signomial programming problems.Section 3 reviews the theoretical and algorithmic developments of mixed-integer nonlinear programming problems.Conclusions are made in Section 4. The first class of SP approaches includes various heuristic techniques such as genetic algorithms, simulated annealing, tabu search, ant colony optimization, and particle swarm optimization.Although the heuristic methods have the advantage of easy implementation and offer a better potential for complex problems, the obtained solution is not guaranteed to be a globally optimal solution.The second class of SP approaches is the deterministic method.For example, Maranas and Floudas 50 , Floudas and Pardalos 51 , Maranas and Floudas 22 , and Floudas 19-21 developed global optimization algorithms for solving SP problems based on the exponential variable transformation, the convex relaxation, and the branch and bound type algorithm.These methods transform the original nonconvex problem into a convex problem and then solve it to obtain the global optimum.The use of the logarithmic/exponential transformation in global optimization algorithms on SP problems restricts these exponential-based methods to handle the problems with strictly positive variables.Although positive variables are employed frequently to represent engineering and scientific systems, it is also common to introduce nonpositive variables in modeling the management problems or the system behavior, such as investment decisions, stresses, temperatures, electrical currents, velocities, and accelerations.For treating free variable x, Pörn et al. 52 suggested a simple translation, x τ e X .However, inserting the transformed result into the original signomial term will bring additional signomial terms and therefore increase the computation burden.Tsai et al. 11 proposed an approach to treat zero boundary signomial discrete programming problems and suggested some convexification rules.Li and Tsai 53 , Tsai and Lin 54-56 , Tsai et al. 57 , Tsai 58 , and Li and Lu 59 applied convexification strategies and piecewise linearization techniques to solve SP problems with free discrete/continuous variables.However, the optimal solution obtained is an approximate solution by the piecewise linearization approach.Lin and Tsai 60 presented a generalized method to solve signomial discrete programming problems with free variables for finding exactly alternative optima.Tsai and Lin 61 also integrated the convexification techniques and the bounding schemes to solve a posynomial geometric program with separable functions for finding a global optimal solution efficiently.

Signomial Programming
Convexification strategies for signomial terms are important techniques in global optimization for SP problems.With different convexification approaches, an SP problem can be reformulated into another convex program solvable to obtain an approximately global optimum.For solving SP problems, P örn et al. 52 integrated the exponential transformation and piecewise linear approximations for reformulating nonconvex signomial problems.The results were extended by Bj örk 62 , Björk et al. 63 , and Pörn et al. 64 , by including certain power transformations for convexification of nonconvex signomial terms.They discussed that the right choice of transformation for convexifying nonconvex signomial terms has a clear influence on the efficiency of the optimization approach.The concept of power convex functions is introduced to improve the solution efficiency for certain SP problems.T. Westerlund and J. Westerlund 65 proposed the generalized geometric programming extended cutting plane GGPECP algorithm for nonconvex optimization problems by using the cutting plane and transformation techniques.The GGPECP algorithm was described in more detail in Westerlund . ., α m > 0 and α m 1 , . . ., α n < 0 , they suggested either the exponential transformation ET or the power convex transformation PCT is applied based on the characteristics of the problems.The ET strategy and the PCT strategy are described as follows 63 .
The ET strategy: The PCT strategy: this technique aims at constructing 1-convex signomial terms.First, transform all variables with positive exponents by an inverse transformation IT , x X −1 , except the one with the greatest exponent denoted as α max .Let S be defined as then transform the variable with the exponent α max to that with the exponent S 1.If α max > S 1, then change one of the ITs, with the exponent α j to x j X −τ j , where τ > 1 so that α max S 1 τ − 1 α j .
Since some negative signomial terms may exist in SP problems, they suggested the potential transformation PT for a negative signomial term . ., α m > 0 and α m 1 , . . ., α n < 0 expressed as follows.
The PT strategy:

2.2
In addition to convexification strategies, convex envelopes and convex underestimators of nonconvex functions are frequently applied in global optimization algorithms such as the αBB algorithm 19, 82, 83 to underestimate the nonconvex functions.A good convex underestimator should be as tight as possible and contain minimal number of new variables and constraints thus to improve the computational effect of processing a node in a branchbound tree 40 .
Tawarmalani and Sahinidis 84 developed the convex envelope and concave envelope for x/y over a unit hypercube, proposed a semidefinite relaxation of x/y, and suggested convex envelopes for functions of the form f x y 2 and f x /y.Ryoo and Sahinidis 85 studied the use of arithmetic intervals, recursive arithmetic intervals, logarithmic transformation, and exponential transformation for multilinear functions.Tawarmalani et al. 86 studied the role of disaggregation in leading to tighter linear programming relaxations.Tawarmalani and Sahinidis 38 introduced the convex extensions for lower semicontinuous functions, proposed a technique for constructing convex envelopes for nonlinear functions, and studied the maximum separation distance for functions such as x/y.Tawarmalani et al. 87 studied 0-1 hyperbolic programs and developed eight mixed-integer convex reformulations.Liberti and Pantelides 88 proposed a nonlinear continuous and differentiable convex envelope for monomials of odd degree, derived its linear relaxation, and compared to other relaxations.Bj örk et al. 63 studied convexifications for signomial terms, introduced properties of power convex functions, compared the effect of the convexification schemes for heat exchanger network problems, and studied quasiconvex convexifications.Meyer and Floudas 89 studied trilinear monomials with positive or negative domains, derived explicit expressions for the facets of the convex and concave envelopes, and showed that these outperform the previously proposed relaxations based on arithmetic intervals or recursive arithmetic intervals.Meyer and Floudas 90 presented explicit expressions for the facets of convex and concave envelopes of trilinear monomials with mixed-sign domains.Tardella 91 studied the class of functions whose convex envelope on a polyhedron coincides with the convex envelope based on the polyhedron vertices and proved important conditions for a vertex polyhedral convex envelope.Meyer and Floudas 92 described the structure of the polyhedral convex envelopes of edge-concave functions over polyhedral domains using geometric arguments and proposed an algorithm for computing the facets of the convex envelopes.
Caratzoulas and Floudas 93 proposed novel convex underestimators for trigonometric functions, which are trigonometric functions themselves.Akrotirianakis and Floudas 94, 95 introduced a new class of convex underestimators for twice continuously differentiable nonlinear programs, studied their theoretical properties, and proved that the resulting convex relaxation is improved compared to the αBB one.Meyer and Floudas 90 proposed two new classes of convex underestimators for general C 2 nonlinear programs, which combine the αBB underestimators within a piecewise quadratic perturbation, derived properties for the smoothness of the convex underestimators, and showed the improvements over the classical αBB convex underestimators for box-constrained optimization problems.
Three popular convex underestimation methods, arithmetic intervals AIs 96 , recursive arithmetic intervals rAIs 50, 85, 96 , and explicit facets EFs for convex envelopes of trilinear monomials 89, 90 , are effective to underestimate a trilinear term x 1 x 2 x 3 for x i to be bounded variables.However, these existing methods have difficulty to treat a posynomial function.According to Ryoo and Sahinidis 85 , for underestimating a multilinear function x 1 x 2 • • • x n with n variables, the AI scheme needs to use n−1 k 2 Θ n k k n/2 i 1 n 2i linear constraints maximally.Θ k denotes the number of linear functions that the AI scheme generates to lower bound k-cross-product terms, k 2, 3, . . ., n − 1.Since the number of linear constraints of convex envelopes for a multilinear function with n variables grows doubly exponentially in n, AI bounding scheme may only treat n ≤ 3 cases.It is more difficult for AI to treat a posynomial function for n > 3 cases.Moreover, applying rAI scheme to underestimate a multilinear function x 1 x 2 • • • x n needs to use the maximum of exponentially many 2 n−1 linear inequalities.Therefore, the rAI bounding scheme has difficulty to treat posynomial functions as well as the AI scheme.EF 89, 90 provided the explicit facets of the convex and concave envelopes of trilinear monomials and demonstrated that these result in tighter bounds than the AI and rAI techniques.An important difference between EF and other bounding schemes is that these explicit facets are linear cuts, which were proven to define the convex envelope.Explicit facets EFs of the convex envelope are effective in treating general trilinear monomials, but the derivation of explicit facets for the convex envelope of general multilinear monomials and signomials is an open problem.Li et al. 97 and Lu et al. 98 proposed a novel method for the convex relaxation of posynomial functions.The approach is different from the work of Maranas and Floudas 50 , which provided an alternative way of generating convex underestimators for generalized geometric programming problems via the exponential transformation and linear underestimation of the concave terms.Applications of this approach include the area of process synthesis and design of separations, phase equilibrium, nonisothermal complex reactor networks, and molecular conformation problems e.g., 99-101 .

Mixed-Integer Nonlinear Programming
Mixed-integer nonlinear programming MINLP problems involving both continuous and discrete variables arise in many applications of engineering design, chemical engineering, operations research, and management.Biegler and Grossmann 102 provided a retrospective on optimization techniques that have been applied in process systems engineering.They indicated that design and synthesis problems have been dominated by NLP and MINLP models.With the increasing reliance on modeling optimization problems in practical problems, a number of theoretical and algorithmic contributions of MINLP have been proposed.Many deterministic methods for convex MINLP problems have been reviewed by Biegler and Grossmann 102 , Grossmann 103 , and Grossmann and Biegler 104 .The methods include branch-and-bound BB 53, 72, 77 , generalized benders decomposition GBD 76 , outerapproximation OA 73, 74, 79 , extended cutting plane method ECP 81 , and generalized disjunctive programming GDP 78 .The BB method can find the global solution only when each subproblem can be solved to global optimality.The GBD method, the OA method, and the ECP method cannot solve optimization problems with nonconvex constraints or nonconvex objective functions because the subproblems may not have a unique optimum in the solution process.The GDP models address discrete/continuous optimization problems that involve disjunctions with nonlinear inequalities and logic propositions.The objective functions and the constraints in the GDP problem are assumed to be convex and bounded 56 .
For deterministic optimization methods, these optimization problems are characterized by the convexity of the feasible domain or the objective function and may involve continuous and/or discrete variables.Although continuous and discrete optimization problems constitute two classes of global optimization problems, they primarily differ in the presence or absence of convexity rather than other features.Since the convexity of the objective function or the feasible domain is very important, understanding how to convexify the nonconvex parts is an essential area of research.As long as the formulated problem is a convex problem, efficient numerical methods are available to treat the optimization problem.However, optimization problems often include nonconvex functions that cannot be dealt with by the standard local optimization techniques to guarantee global optimality efficiently.For solving nonconvex or large-scale optimization problems, deterministic methods may not be easy to derive an optimal solution within reasonable time due to the high complexity of the problem.
Sherali et al. 105 presented an extension of the reformulation linearization technique RLT that is designed to exploit special structures and explored the strengthening of the RLT constraints through conditional logical expressions.Sinha et al. 106 studied a solvent design problem that is constructed as a nonconvex MINLP problem.They identified the sources of nonconvexities in the properties and solubility parameter design constraints and proposed linear underestimators based on a multilevel representation approach for the functions.A reduced space branch-and-bound global optimization algorithm was then presented for solving a single component blanket wash design problem.P örn et al. 52 introduced different convexification strategies for nonconvex MINLP problems with both posynomial and negative binomial terms in the constraints.Harjunkoski et al. 107 studied the trim loss minimization problem for the paper converting industry and formulated the model as a nonconvex MINLP.They also proposed transformations for the bilinear terms based on linear representations and convex expressions.P örn and Westerlund 108 proposed a cutting plane method for addressing global MINLP problems with pseudoconvex objective function and constraints and tested the proposed method on several benchmark problems arising in process synthesis and scheduling applications.Parthasarathy and El-Halwagi 109 studied a systematic framework for the optimal design of condensation, which is an important technology for volatile organic compounds, and formulated the problem as a nonconvex MINLP model.They also proposed an iterative global optimization approach based on physical insights and active constraint principles that allow for decomposition and efficient solution and applied it to a case study for the manufacture of adhesive tapes.Adjiman et al. 82,83,110,111 proposed two global optimization approaches, SMIN-αBB and GMIN-αBB, for nonconvex MINLP problems based on the concept of branch-and-bound.These two approaches rely on optimization or interval-based variable-bound updates to enhance efficiency.Although one possible approach to circumvent nonconvexities in nonlinear optimization models is reformulation, for instance, using the exponential transformation to treat the generalized geometric programming problems in which a signomial term x α 1 x β 2 is transferred into an exponential term e α ln x 1 β ln x 2 , the exponential transformation technique can only be applied to strictly positive variables and is thus unable to deal with nonconvex problems with free variables.Tsai et al. 11 proposed an approach to treat zero boundary optimization problems and suggested some convexification rules for the signomial terms with only three nonnegative discrete/integer variables.Bj örk and Westerlund 112 studied the global optimization of heat exchanger network synthesis through the simplified superstructure representation that allows only series and parallel schemes and applied convexification approaches for signomials by piecewise linear approximations.They also formulated convex MINLP lower bounding models using the Patterson formula for the log mean temperature difference considering both isothermal and nonisothermal mixing.Ostrovsky et al. 113 studied nonconvex MINLP models in which most variables are in the nonconvex terms and the number of linear constraints is much larger than the number of nonlinear constraints for solvent design and recovery problems.The work presents a tailored branch-and-bound approach using linear underestimators for tree functions based on a multilevel function representation and shows that there is a significant reduction in the branching variable space.Tawarmalani and Sahinidis 114 developed a branch and bound framework for the global optimization of MINLP problems.The framework involves novel linear relaxation schemes, a Lagrangian/linear duality-based theory for domain and range reduction, and branching strategies that guarantee finiteness of the solution sequence for certain classes of problems.They also discuss implementation issues and present computational results with a variety of benchmark problems.Kesavan et al. 115 presented outer-approximation algorithms for finding an optimal solution of a separable nonconvex MINLP program.Emet and Westerlund 116 conducted a computational comparison of solving a cyclic chromatographic separation problem using MINLP methods and reported that the extended cutting plane method compares favourably against traditional outerapproximation and branch-and-bound methods.A review of the recent advances in MINLP optimization of planning and design problems in the process industry was presented by Kallrath 117 .Tawarmalani and Sahinidis 118 introduced a polyhedral branch-and-cut approach in global optimization.Their algorithm exploits convexity in order to generate the polyhedral cuts and relaxations for multivariate nonconvex problems.Meyer and Floudas 119 studied superstructures of pooling networks, which are important to the petrochemical, chemical, and wastewater treatment industries, and formulated this generalized pooling problem as a nonconvex MINLP problem that involves many bilinear terms in the constraint functions.They proposed a global optimization algorithm based on a novel piecewise linear reformulation-linearization technique RLT formulation.Karuppiah and Grossmann 120 addressed the problem of optimal synthesis of an integrated water system, where water using processes and water treatment operations are jointly considered.The designed MINLP model was solved with a new deterministic spatial branch and contract algorithm, in which piecewise under-and overestimators are used for constructing the relaxations at each node.Bergamini et al. 121 formulated an MINLP model for the global optimization of heat exchanger networks and presented a new solution methodology that is based on outer-approximation and utilizes piecewise underestimation.Rigorous constraints obtained from physical insights are also included in the formulation, and the authors reported computationally efficient global solutions for problems with up to nine process streams.Tsai and Lin 54, 56 proposed a method for solving a signomial MINLP problem with free variables by the convexification strategies and piecewise linearization techniques.However, the optimal solution obtained is an approximate solution by the piecewise linearization approach.Karuppiah et al. 122 presented an outer-approximation algorithm to globally solve a nonconvex MINLP formulation that corresponds to the continuous time scheduling of refinery crude oil operations.The solution procedure relies on effective mixed-integer linear relaxations that benefit from additional cuts derived after spatially decomposing the network.Foteinou et al. 123 presented a mixed-integer optimization framework for the synthesis and analysis of regulatory networks.Their approach integrates prior biological knowledge regarding interactions between genes and corresponding transcription factors, in an effort to minimize the complexity of the problem.Misener et al. 124 proposed an extended pooling problem to maximize the profit of blending reformulated gasoline on a predetermined network structure of feed stocks, intermediate storage tanks, and gasoline products subject to applicable environmental standards.They formulated the problem as a nonconvex MINLP model due to the presence of bilinear, polynomial, and fractional power terms.A mixedinteger linear programming relaxation of the extended pooling problem is proposed for several small-to large-scale test cases.Misener et al. 125 introduced a formulation for the piecewise linear relaxation of bilinear functions with a logarithmic number of binary variables and computationally compared their performance of this new formulation to the best performing piecewise relaxations with a linear number of binary variables.They also unified the new formulation into the computational tool APOGEE that globally optimizes standard, generalized, and extended pooling problems.Westerlund et al. 126 considered some special but fundamental issues related to convex relaxation techniques in nonconvex MINLP optimization, especially for optimization problems including nonconvex inequality constraints and their relaxations.
The alternative global optima of an MINLP problem can be found if more than one solution satisfies the same optimal value of the objective function.In practice, alternative optima are useful because they allow the decision maker to choose from many solutions without experiencing any deterioration in the objective function.For the case involving only 0-1 variables, Balas and Jeroslow 127 introduced the well-known binary cut with only one constraint and no additional variables.Duran and Grossmann 73 used this binary cut in their OA algorithm to exclude binary combinations.Tawarmalani and Sahinidis 38 mentioned that BARON can identify the K best solutions for a mixed-integer nonlinear program, where K is an option specified by the user.Tsai et al. 57 proposed a general integer cut to identify all alternative optimal solutions of a general integer linear programming problem.Lin and Tsai 60 proposed a generalized method to find multiple optimal solutions of an MINLP problem with free variables by means of variable substitution and convexification strategies.The problem is first converted into another convex MINLP problem solvable to obtain an exactly global optimum.Then, a general cut is utilized to exclude the previous solution and an algorithm is developed to locate all alternative optimal solutions.

Conclusions
Given the rapid advances in computing technology over the past decades, large optimization theories and algorithms have been proposed to solve various real-world engineering and management problems.Therefore, to give a systematic overview of the extant literature is a challenge and motivates this study, particularly for that the field of optimization has grown and evolved rapidly.This work first reviewed methods for continuous variable optimization and survey advances in signomial programming.Then, mixed-integer nonlinear programming methods for optimization problems with discrete components were introduced.Contributions related to theoretical and algorithmic developments, formulations, and applications for these two classes of optimization problems were also discussed.
Although deterministic approaches take advantage of analytical properties of the problem to generate a sequence of points that converge to a global solution, heuristic approaches have been found to be more flexible and efficient than deterministic approaches.For solving nonconvex or large-scale optimization problems, deterministic methods may not be easy to derive a globally optimal solution within reasonable time due to the high complexity of the problem.Heuristic approaches therefore are presented to reduce the computational time of solving an optimization problem, but the obtained solution is not guaranteed to be a feasible or globally optimal solution.These two types of optimization methods have different pros and cons.Therefore, integrating deterministic and heuristic approaches may be a good way of solving large-scale optimization problems for finding a global optimum.It is hoped that this paper will stimulate further research on developing more advanced deterministic and heuristic methods to enhance the computational efficiency of finding a globally optimal solution for various real application problems.
Floudas and Pardalos 32 studied optimization approaches in the fields of computational chemistry and molecular biology.Laguna and González-Velarde 33 discussed advanced computing tools for tackling various challenging optimization problems.Papalambros and Wilde 34 investigated the principles and practice of optimal engineering design.Atkinson et al. 35 gave a detailed discussion on optimum experimental design.Edgar et al. 36 explored the optimization methods of chemical processes.Pardalos and Romeijn 37 provided a more complete and broad spectrum of approaches including deterministic and heuristic techniques for dealing with global optimization problems.Tawarmalani and Sahinidis 38 provided an insightful and comprehensive treatment of convexification and global optimization of continuous and mixed-integer nonlinear programs.Hadjisavvas et al. 39 investigated generalized convexity and generalized monotonicity and offered an advanced and broad overview of the state of the field.Moreover, Floudas et al. 40 presented an overview of the research progress in optimization during 1998-2003, including the deterministic global optimization advances in mixed-integer nonlinear programming and related applications.Pintér 41 illustrated the applicability of global optimization modeling techniques and solution strategies to real-world problems such as agroecosystem management, assembly line design, bioinformatics, biophysics, cellular mobile network design, chemical product design, composite structure design, controller design for induction motors, electrical engineering design, feeding strategies in animal husbandry, the inverse position problem in kinematics, laser design, radiotherapy planning, robot design, and satellite data analysis.Mishra and Giorgi 42 presented results on invex function, and their properties in smooth and nonsmooth cases, pseudolinearity, and etapseudolinearity.Mishra et al. 43 discussed the Kuhn-Tucker optimality, Karush-Kuhn-Tucker necessary and sufficient optimality conditions in presence of various types of generalized convexity assumptions.Floudas and Gounaris 44 also presented an overview of the research progress in deterministic global optimization during the last decade 1998-2008 .

Figure 1 :
Figure 1: Overview of problem types related to optimization problems.
66 .Lundell and Westerlund 67 and Lundell et al. 68 combined the GGPECP algorithm with an optimization framework for the transformations used to convexify the signomial terms into a signomial global optimization algorithm.The signomial global optimization algorithm was further extended by Lundell and Westerlund 69, 70 .Lin and Tsai 71 and Tsai and Lin 56 also presented similar reformulation and range reduction techniques to enhance the computational efficiency for solving SP problems.For solving an SP problem, the above-mentioned convexification techniques are used to reformulate the original SP problem into a convex and underestimating problem solvable by a standard mixed-integer nonlinear programming MINLP solver 72-81 .Different transformations for positive and negative signomial terms have been proposed and discussed by Bj örk et al. 63 , Westerlund 66 , Lundell and Westerlund 67 , Pörn et al. 64 , Lundell et al. 68 , and Lundell and Westerlund 69, 70 .For a positive signomial term et al. 26 investigated decision support systems and techniques for solving molecular structures, queuing systems, image reconstruction, location analysis, and process network synthesis problems.Migdalas et al. 27 presented multilevel optimization algorithms and their applications.Mistakidis and Stavroulakis 28 studied engineering applications of the finite element method.De Leone et al. 29 proposed various interesting applications of highperformance software for nonlinear optimization.Hendrix 30 utilized global optimization techniques on environmental management, geometric design, robust product design, and parameter estimation.Corliss and Kearfott 31 presented a rigorous global search method on industrial applications.

Signomial programming SP is an optimization technique for solving a class of nonconvex nonlinear programming problems. Although SP problems occur frequently in engineering and management science, SP problems with nonconvex functions are still difficult to be solved to obtain a global optimum. The term geometric program GP was introduced by Duffin et al. 7 since the analysis of geometric programs relied heavily upon geometric
arithmetic mean inequality.The early work by Duffin and Peterson 45 solved the posynomial geometric program analytically via the dual problem.Then, Duffin 46 developed a numerical method for GPs based on solving a sequence of linear programs 47 .For solving SP problems, Duffin and Peterson 48 reformulated an SP problem as a geometric program with reversed constraints.The reversed constraints give rise to a nonconvex feasible region that the local minima for SP problems are not guaranteed to be global minima 49 .The developed methods for SP can be divided into two approaches.