scholarly journals A Review of Piecewise Linearization Methods

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Ming-Hua Lin ◽  
John Gunnar Carlsson ◽  
Dongdong Ge ◽  
Jianming Shi ◽  
Jung-Fa Tsai
Keyword(s):  
Nonlinear Programming ◽  
Programming Problem ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Convex Programming Problem ◽  
Nonlinear Programming Problem ◽  
Global Optimal Solution ◽  
Piecewise Linearization ◽  
Linearization Methods ◽  
Global Optimal

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.

Reactive Power Optimization for Distribution Network with Distributed Generators Based on Semi-Definite Programming

2014 ◽  
Vol 1070-1072 ◽  
pp. 809-814
Author(s):  
Lei Dong ◽  
Ai Zhong Tian ◽  
Tian Jiao Pu ◽  
Zheng Fan ◽  
Ting Yu
Keyword(s):  
Mathematical Model ◽  
Reactive Power ◽  
Power Optimization ◽  
Distribution Network ◽  
Optimal Solution ◽  
Mixed Integer ◽  
Reactive Power Optimization ◽  
Global Optimal Solution ◽  
Distributed Generators ◽  
Global Optimal

Reactive power optimization for distribution network with distributed generators is a complicated nonconvex nonlinear mixed integer programming problem. This paper built a mathematical model of reactive power optimization for distribution network and a new method to solve this problem was proposed based on semi-definite programming. The original mathematical model was transformed and relaxed into a convex SDP model, to guarantee the global optimal solution within the polynomial times. Then the model was extended to a mixed integer semi-definite programming model with discrete variables when considering discrete compensation equipment such as capacitor banks. Global optimal solution of this model can be obtained by cutting plane method and branch and bound method. Numerical tests on the modified IEEE 33-bus system show this method is exact and can be solved efficiently.

scholarly journals A Linearization to the Sum of Linear Ratios Programming Problem

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1004
Author(s):  
Mojtaba Borza ◽  
Azmin Sham Rambely
Keyword(s):  
Programming Problem ◽  
Computational Cost ◽  
Iterative Algorithms ◽  
Optimal Solution ◽  
Optimal Solutions ◽  
Global Optimal Solution ◽  
Straightforward Method ◽  
Global Optimal ◽  
High Computational Cost ◽  
Real World Problems

Optimizing the sum of linear fractional functions over a set of linear inequalities (S-LFP) has been considered by many researchers due to the fact that there are a number of real-world problems which are modelled mathematically as S-LFP problems. Solving the S-LFP is not easy in practice since the problem may have several local optimal solutions which makes the structure complex. To our knowledge, existing methods dealing with S-LFP are iterative algorithms that are based on branch and bound algorithms. Using these methods requires high computational cost and time. In this paper, we present a non-iterative and straightforward method with less computational expenses to deal with S-LFP. In the method, a new S-LFP is constructed based on the membership functions of the objectives multiplied by suitable weights. This new problem is then changed into a linear programming problem (LPP) using variable transformations. It was proven that the optimal solution of the LPP becomes the global optimal solution for the S-LFP. Numerical examples are given to illustrate the method.

A Deterministic Method for a Class of Fractional Programming Problems with Coefficients

2011 ◽  
Vol 467-469 ◽  
pp. 531-536
Author(s):  
Jing Ben Yin ◽  
Kun Li
Keyword(s):  
Optimization Problems ◽  
Optimal Solution ◽  
Deterministic Algorithm ◽  
Linear Relaxation ◽  
Convex Programming Problem ◽  
Global Optimal Solution ◽  
Deterministic Method ◽  
Equivalent Problem ◽  
Linear Programming Problems ◽  
Global Optimal

The sum of linear fractional functions problem has attracted the interest of researchers and practitioners for a number of years. Since these types of optimization problems are non-convex, various specialized algorithms have been proposed for globally solving these problems. However, these algorithms are only for the case that sum of linear ratios problem without coefficients, and may be difficult to be solved. In this paper, a deterministic algorithm is proposed for globally solving the sum of linear fractional functions problem with coefficients. By utilizing an equivalent problem and linear relaxation technique, the initial non-convex programming problem is reduced to a sequence of linear relaxation programming problems. The proposed algorithm is convergent to the global optimal solution by means of the subsequent solutions of a series of linear programming problems.

scholarly journals Semiconductor Yield Forecasting Using Quadratic-Programming-Based Fuzzy Collaborative Intelligence Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Toly Chen ◽  
Yu-Cheng Wang
Keyword(s):  
Nonlinear Programming ◽  
Quadratic Programming ◽  
Optimal Solution ◽  
Global Optimal Solution ◽  
Expert Opinions ◽  
Forecasting Methods ◽  
Global Optimal ◽  
Yield Forecasting ◽  
Collaborative Intelligence ◽  
Intelligence Approach

Several recent studies have proposed fuzzy collaborative forecasting methods for semiconductor yield forecasting. These methods establish nonlinear programming (NLP) models to consider the opinions of experts and generate fuzzy yield forecasts. Such a practice cannot distinguish between the different expert opinions and can not easily find the global optimal solution. In order to solve some problems and to improve the performance of semiconductor yield forecasting, this study proposes a quadratic-programming- (QP-) based fuzzy collaborative intelligence approach.

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