Ming-Hua Lin

On generalized geometric programming problems with non-positive variables

Generalized geometric programming (GGP) problems occur frequently in engineering design and management. Some exponential-based decomposition methods have been developed for solving global optimization of GGP problems. However, the use of logarithmic/exponential transformations restricts these methods to handle the problems with strictly positive variables. This paper proposes a technique for treating non-positive variables with integer powers in GGP problems. By means of variable transformation, the GGP problem with non-positive variables can be equivalently solved with another one having positive variables. In addition, we present some computationally efficient convexification rules for signomial terms to enhance the efficiency of the optimization approach. Numerical examples are presented to demonstrate the usefulness of the proposed method in GGP problems with non-positive variables.

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.

Finding multiple solutions to general integer linear programs

Integer linear programming (ILP) problems occur frequently in many applications. In practice, alternative optima are useful since they allow the decision maker to choose from multiple solutions without experiencing any deterioration in the objective function. This study proposes a general integer cut to exclude the previous solution and presents an algorithm to identify all alternative optimal solutions of an ILP problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.

Global optimization of signomial mixed-integer nonlinear programming problems with free variables

Mixed-integer nonlinear programming (MINLP) problems involving general constraints and objective functions with continuous and integer variables occur frequently in engineering design, chemical process industry and management. Although many optimization approaches have been developed for MINLP problems, these methods can only handle signomial terms with positive variables or find a local solution. Therefore, this study proposes a novel method for solving a signomial MINLP problem with free variables to obtain a global optimal solution. The signomial MINLP problem is first transformed into another one containing only positive variables. Then the transformed problem is reformulated as a convex mixed-integer program by the convexification strategies and piecewise linearization techniques. A global optimum of the signomial MINLP problem can finally be found within the tolerable error. Numerical examples are also presented to demonstrate the effectiveness of the proposed method.

Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems

Many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations. Since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem, this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency. In the proposed algorithm, the non-convex geometric programming problem is first converted into a convex mixed-integer nonlinear programming problem by convexification and piecewise linearization techniques. Then, an optimization-based approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process, therefore decreasing the required CPU time. Several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.

A Review of Piecewise Linearization Methods

Various optimization problems in engineering and management are formulated as nonlinear programming problems. Because of the nonconvexity nature of this kind of problems, no efficient approach is available to derive the global optimum of the problems. How to locate a global optimal solution of a nonlinear programming problem is an important issue in optimization theory. In the last few decades, piecewise linearization methods have been widely applied to convert a nonlinear programming problem into a linear programming problem or a mixed-integer convex programming problem for obtaining an approximated global optimal solution. In the transformation process, extra binary variables, continuous variables, and constraints are introduced to reformulate the original problem. These extra variables and constraints mainly determine the solution efficiency of the converted problem. This study therefore provides a review of piecewise linearization methods and analyzes the computational efficiency of various piecewise linearization methods.

An optimization approach for solving signomial discrete programming problems with free variables

Signomial discrete programming (SDP) problems occur frequently in engineering design. This paper proposes a generalized method to solve SDP problems with free variables. An SDP problem with free variables is first converted into another one containing non-negative variables, and then various non-convex signomial terms are transformed such that the original SDP problem becomes a convex integer program solvable to obtain a globally optimal solution. Compared with current SDP methods, the proposed method is capable of dealing with free variables of an SDP problem and is guaranteed to converge to a global optimum. In addition, several computationally efficient convexification rules for signomial terms are presented to enhance the efficiency of the optimization approach. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method.

FLEXOR: a flexible localization scheme based on RFID

There are many localization schemes used for the indoor or outdoor applications, such as the GPS, RADAR etc However, the accuracy of indoor localization scheme is easy to be influenced because of the obstacles and the environment interference LANDMARC which is proposed by Lionel et al uses the RFID tags to reduce the influence of the interference This paper proposes an improved localization scheme, FLEXOR, which divides the localization area into cells to reduce computational overhead and provide two localization modes: region mode and coordinate mode In the performance evaluations, FLEXOR has been proved to have the advantages of fast localization, flexibility, and it also provides the high localization accuracy as LANDMARC.

An Efficient Global Approach for Posynomial Geometric Programming Problems

Aposynomial geometric programming problem is composed of a posynomial being minimized in the objective function subject to posynomial constraints. This study proposes an efficient method to solve a posynomial geometric program with separable functions. Power transformations and exponential transformations are utilized to convexify and underestimate posynomial terms. The inverse transformation functions of decision variables generated in the convexification process are approximated by superior piecewise linear functions. The original program therefore can be converted into a convex mixed-integer nonlinear program solvable to obtain a global optimum. Several numerical experiments are presented to investigate the impact of different convexification strategies on the obtained approximate solution and to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.

Finding all solutions of systems of nonlinear equations with free variables

Systems of nonlinear equations often represent mathematical models in engineering design. This study proposes a novel method for finding all solutions of systems of nonlinear equations with free variables. The original problem is first transformed into a global optimization problem whose multiple global minima with a zero objective value correspond to all solutions of the original problem. Then, by using variable substitution on free variables and applying convexification strategies and piecewise linearization techniques on nonconvex functions, the transformed optimization problem is reformulated as a convex mixed-integer program solvable to reach an approximately global optimum. An algorithm is developed to find all solutions of the reformulated problem. Numerical examples in real applications are presented to demonstrate the usefulness of the proposed method in engineering design.