An Efficient Global Approach for Posynomial Geometric Programming Problems

Abstract

A posynomial 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.

A Review of Piecewise Linearization Methods

“…According to the above discussions, Method 5 is more computationally efficient than the other four methods. Experiment results from the literature [39,48,49] also support the statement.…”

“…For evaluating the error of piecewise linear approximation, Tsai and Lin [49,52] and Lin and Tsai [53] utilized the expression | ( ) − ( ( ))| to estimate the error indicated in Figure 2. If ( ) is the objective function, ( ) < 0 is the th constraint, and * is the solution derived from the transformed program, then the linearization does not require to be refined until | ( * ) − ( ( * ))| ≤ 1 and Max ( ( * )) ≤ 2 , where | ( * ) − ( ( * ))| is the evaluated error in objective, 1 is the optimality tolerance, ( * ) is the error in the th constraint, and 2 is the feasibility tolerance.…”

“…According to the deterministic optimization methods for solving nonconvex nonlinear problems [29,33,38,39,48,49,[53][54][55][56], the inverse or logarithmic transformation is required to be approximated by the piecewise linearization function. For example, the function = ln or = −1 is required to be piecewisely linearized by using an appropriate breakpoint selection strategy, if a new break point is added at the midpoint of each interval of existing break points or at the point with largest approximation error, the number of line segments becomes double in each iteration.…”

“…Vielma et al [39] presented a note on Li et al 's paper and showed that two representations for piecewise linear functions introduced by Li et al [48] are both theoretically and computationally inferior to standard formulations for piecewise linear functions. Tsai and Lin [49] applied the Vielma et al [39] techniques to express a piecewise linear function for solving a posynomial optimization problem. Croxton et al [31] indicated that most models of expressing piecewise linear functions are equivalent to each other.…”

Engineering Design by Geometric Programming

“…GP has been applied in many fields of applications including analog/digital circuit design [2][3][4][5][6][7][8], chemical engineering [9][10][11], mechanical engineering [12][13][14][15][16][17][18][19], power control [20], and communication network systems [21][22][23].…”