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Convexification Strategies for Signomial Programming Problems

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### Convexification Strategies for Signomial Programming Problems

Jung-Fa Tsai

Department of Business Management

National Taipei University of Technology

Outline

- Introduction & Problem Formulation
- Conventional Approach
- Convexification Strategies
- Piecewise Linearization Techniques
- Example
- Advantages & Future Directions

Introduction

- The signomial programming (SP) problem occurs frequently in engineering and management sciences.
- Most of current methods can only obtain local solutions. There is no efficient method for obtaining the global optimum of a SP problem.
- Here we solve a SP problem based on the proposed convexification strategies and piecewise linearization techniques.

Minimize convex function

subject to {convex set}.

Maximize concave function

subject to {convex set}.

How to obtain a global solution?Conventional Approach -Exponential Transformation

Denote for xi > 0, the SP program can be rewritten as the following program with exponential form:

Minimize

subject to

Example

If , z can be transformed into the exponential form

Three concave terms and

have to be linearized.

Lemma 1 For a twice-differentiable function

, , denote is the Hessian matrix of . The determinant of can be expressed as

.

Proposition 1A twice-differentiable function is convex for , , .

Proposed Method-Convexification(1)Proposition 2A twice-differentiable function is convex for , , , , .

EX: A function for is convex when or and is concave when .

Convexification(2)Theorem 1 For all , a term where for all i and for can be convexified as follows.

(i) ,

(ii) for ,

(iii) for ,

where is a piecewise linearization function of a concave term .

Convexification(3)Theorem 2 For all , a term where for all i and for all can be convexified as follows.

(i) , ,

(ii) for ,

(iii) for ,

where is a piecewise linearization function of a concave term .

Convexification(4)where is a linearization function of , and , are the break points of , ; and

are the slopes of line segments between and ,

for j=1,2,…,m-1.

Piecewise LinearizationProposition 3 A concave function can be piecewisely approximated as:

Example- Container Loading Problem

Minimize

subject to

1. All boxes are non-overlapping,

2. All boxes are within the ranges of x, y, z.

Transformed Program

Minimize

subject to

1. All boxes are non-overlapping,

2. All boxes are within the ranges of x, y, z.

Conclusions

- Advantage:
- The proposed method can solve a SP program to find the solution which can be as close as possible to the global optimum, instead of obtaining a local optimum;
- The number of concave terms required to be linearized by the proposed method is fewer than that by the other global optimization techniques.(i.e.,more computationally efficient).
- Future directions: Develop definite convexification rules, distributed-computation algorithms, integrate with heuristic approaches, and apply the techniques to the real world problems.

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