Convexification Strategies for Signomial Programming Problems

1. Convexification Strategies for Signomial Programming Problems Jung-Fa Tsai Department of Business Management National Taipei University of Technology

2. Outline • Introduction & Problem Formulation • Conventional Approach • Convexification Strategies • Piecewise Linearization Techniques • Example • Advantages & Future Directions

3. 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.

4. A SP Problem Formulation Minimize subject to

5. Minimize convex function subject to {convex set}. Maximize concave function subject to {convex set}. How to obtain a global solution?

6. Conventional Approach -Exponential Transformation Denote for xi > 0, the SP program can be rewritten as the following program with exponential form: Minimize subject to

7. Example If , z can be transformed into the exponential form Three concave terms and have to be linearized.

8. 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)

9. Proposition 2A twice-differentiable function is convex for , , , , . EX: A function for is convex when or and is concave when . Convexification(2)

10. 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)

11. 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)

12. 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 Linearization Proposition 3 A concave function can be piecewisely approximated as:

13. , are the break points of … x Graphical Illustration

14. Example- Container Loading Problem Minimize subject to 1. All boxes are non-overlapping, 2. All boxes are within the ranges of x, y, z.

15. Transformed Program Minimize subject to 1. All boxes are non-overlapping, 2. All boxes are within the ranges of x, y, z.

16. 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.

17. Thanks for your attention!