Engineering Design by Geometric Programming

. A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions, where all functions are of signomial form. The importance of GP comes from two relatively recent developments: (i) new methods can solve even large-scale GP extremely efficiently and reliably; (ii) a number of practical problems have recently been found to be equivalent to or approximated by GP. This study proposes an optimization approach for solving GP. Our approach is first to convert all signomial terms in GP into convex and concave terms. Then the concave terms are further treated with the proposed piecewise linearization method where only ⌈ log 2 (𝑚 − 1)⌉ binary variables are used. It has the following features: (i) it offers more convenient and efficient means of expressing a piecewise linear function; (ii) fewer 0-1 variables are used; (iii) the computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large. In addition, the engineering design problems are illustrated to evaluate the usefulness of the proposed methods.

Obtaining the optimal solutions for GP is not straightforward because the signomial terms in the objective function and constraints cannot be solved directly.As a result, many approaches have been developed.Coello and Cortés [24] proposed a genetic algorithm with an artificial immune system to solve a GP in engineering optimization.Nevertheless, this method can only obtain the local optima.Horst and Tuy [25] introduced an analytical approach for solving a problem with Lipschitzian objective and constraints.The restriction of this approach is to find the global optimum only if the range of variables can be reduced by analytical techniques.Sherali and Tuncbilek [26] developed a reformulation-linearization technique (RLT) which generates polynomial implied constraints and then linearizes the resulting problem by introducing new variables.Lin and Tsai [14] introduced a generalized method to find multiple optimal solutions of signomial discrete programming problems with free variables.By means of variable substitution and convexification strategies, a signomial discrete programming problem with free variables is first converted into another convex mixed-integer nonlinear programming problem solvable to obtain an exactly global optimum.Tsai [17] proposed a novel method to solve signomial discrete programming problems.The signomial terms are first convexified following the convexification strategies.The original program is then converted into a convex integer program solvable by commercialized packages to obtain globally optimal solutions.Tsai and Lin [19] developed 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 original program therefore can be converted into a convex mixed-integer nonlinear program solvable to obtain a global optimum.
This paper develops an optimization approach for solving GP.Our approach is listed as follows.
(i) Convert all signomial terms into convex and concave ones.
(ii) The concave terms are further treated with the proposed piecewise linearization method where only ⌈log 2 ( − 1)⌉ binary variables are used.
The rest of this paper is organized as follows.Section 2 introduces the proposed methods.Section 3 provides some numerical examples to illustrate the modeling idea and the usefulness of the proposed method.Section 4 gives our conclusions.

Proposed Methods
In this section, the proposed methods will be presented.First, we recall what a signomial function is.A function  : R  ++ → R defined as where  > 0 and   ∈ R for all , is called a monomial function or simply a monomial [1].Note that the exponents   of a monomial can be any real numbers, but the coefficient  must be nonnegative.A sum of monomials, namely, a function of the form where   > 0 and   ∈ R, is called a posynomial function with  terms or simply a posynomial.A signomial is a linear combination of monomials of some positive variables  1 , .The converse of Lemma 1(a) is false.For example, let  = [ 1 0 0 0 0 0 0 0 −1 ]; we have ⟨, ⟩ =  2  1 −  2 3 which is not always nonnegative for all  ∈ R 3 .But Δ 1 = 1 ≥ 0, Δ 2 = 0 ≥ 0, and Δ 3 = 0 ≥ 0. In fact, the converse of Lemma 1(a) is true only for  = 2; see [27, Because   ≤ 0 for all  = 1, 2, . . ., , we know that all eigenvalues of ∇ In addition, it can be verified that Namely, Moreover, the determinant of ∇ 2  2 () can be computed and be shown by induction as We will complete the proof by discussing the following two cases.
From all the previous, the desired result follows.
We want to point out that our results also provide an alternative proof for the main result (Theorem 7) of [28].Indeed, Maranas and Floudas [28] further discuss another condition as follows: to guarantee that  1 defined as in Proposition 3 is a convex function.Our approach can be also employed to verify this fact.To see this, we arrange all powers   in a decreasing order.In other words, without loss of generality, we assume that Notice that condition (15) implies that  1 is positive and all the other  2 , . . .,   are nonpositive with  1 ≥ 1 − ∑  =2   .As mentioned in Proposition 3, we only need to show that the function f1 () = ∏  =1     is convex.By similar arguments as in the proof of Proposition 4, we know that where Δ  denotes the th principal minor of the Hessian matrix of f1 ().From conditions ( 15) and ( 16), it is easily verified that (1 − ∑  =1   ) < 0 for each .It is also not hard to observe that ∏  =1   is positive if  is odd and is negative if  is even.In summary, there holds The above two inequalities yield that Δ  > 0 for each .Thus, following the same arguments as in Proposition 4, we can conclude that  1 is also a convex function under condition (15).
Conventional methods for linearizing a concave function with  break points require  − 1 binary variables.However, when  becomes large, the computation will be very time consuming and may cause a heavy computational burden.An effective piecewise linearization method proposed by Huang [29] is presented in which only ⌈log 2 ( − 1)⌉ binary variables are used.
The following proposition is deduced.
We then have the following theorem.
Step 3. Generate the set of constraints for  and f() as ( 25) and (26), respectively, and solve the resulting problem.

Numerical Examples
In this section, we have conducted some engineering design problems to evaluate the usefulness of the proposed methods.
Example 1.Consider the following engineering design problem of a speed reducer which was proposed by Golinski [37] as in Figure 2.
The objective of this problem is to minimize the weight of the speed reducer while satisfying a number of constraints imposed by gear and shaft design practices.There are seven design variables,  1 (width of the gear face, cm),  2 (teeth module, cm),  3 (number of pinion teeth),  4 (shaft 1 length between bearings, cm),  5 (shaft 2 length between bearings, cm),  6 (diameter of shaft 1, cm), and  7 (diameter of shaft 2, cm).The constraints are characterized by ℎ 1 (upper bound on the bending stress of the gear tooth), ℎ 2 (upper bound on the contact stress of the gear tooth), ℎ 3 , ℎ 4 (upper bounds on the transverse deflection of shafts 1, 2), ℎ 5 , ℎ 6 (upper bounds on the stresses in shafts 1, 2), ℎ 7 , ℎ 8 , ℎ 9 (dimensional restrictions based on space and experience), and ℎ 10 , ℎ 11 (dimensional requirements for shafts based on experience).The problem can be formulated as Algorithm 2. Step 1.The proposed method requires the use of ⌈log 2 (41 − 1)⌉ = 6 binary variables to linearize  1,1 =  −1 1 in a piecewise manner.Compared with the conventional piecewise linearization methods, the number of newly added binary variables for a piecewise linear function with 41 break points is significantly reduced from 40 to 6.

3.575
where Solving the transformed program with MATLAB R13, Table 2 shows the results computed by Coello and Cortés [24] and the proposed method.
Example 2 (see [38]).Consider the design of a welded beam which is to be rigidly attached to a fixed support and is subjected to a load as illustrated in Figure 3.The welded beam is to consist of 1010 steel and is to support a force  of 6000 lb.The length  is assumed to be specified at 14 in.
Three methods (brazed, bolted, and welded connections) of attachment and three materials (steel, brass, and aluminum) are considered.The choices of attachment and materials are to be made, which minimizes cost while satisfying a limit on the allowable bending stress, the buckling load, and the allowable deflection at the load application point.
The various materials have different elastic, shear moduli, and different costs.
The design variables are defined as  1 (a continuous variable for thickness of weld),  2 (a continuous variable for width of beam),  3 (a continuous variable for length of connection),  4 (a continuous variable for thickness of beam),  5 (a continuous variable for diameter of bolts),  6 (a 0-1 variable for brazed connections),  7 (a 0-1 variable for bolted connections),  8 (a 0-1 variable for welded connections),  9 (a 0-1 variable for steel material),  10 (a 0-1 variable for brass material), and  11 (a 0-1 variable for aluminum material).
The problem can be formulated as ( 45)-( 50 where  allowable is the maximum bar bending stress.  is the bar buckling load which is a function of the beam dimensions and material properties. max is the maximum bar deflection. is the Young's modulus, 30 × 10 6 psi.More detailed description including the derivation of   can be found in Reklaitis et al. [38].Following the same procedure as in Example 1, the nonlinear terms  2  3  6 ,  5  7 ,   ,  11 ) = (0.320, 14.109, 5.720, 0.321, 4.992, 0, 0, 1, 0, 0, 1) and the objective value is 1.969 within the tolerance error 0.0001.

Conclusions
This paper proposes an optimization approach for solving geometric programming problems.Our approach is first to convert all signomial terms in GP into convex and concave terms by the proposed methods.Then the concave terms are further treated with the proposed piecewise linearization method where only ⌈log 2 ( − 1)⌉ binary variables are used.It has the following features.(ii) Fewer 0-1 variables are used.
(iii) The computational results show that the proposed method is much more efficient and faster than the conventional one, especially when the number of break points becomes large.
Numerical examples in real applications are illustrated to demonstrate the usefulness of the proposed methods.
(i) It offers more convenient and efficient means of expressing a piecewise linear function.
. .,   .Let  be defined on an open convex set  ⊆ R  and be twice differentiable; it is known that (i)  is convex on  if and only if the Hessian matrix ∇ 2 () is positive semidefinite (p.s.d. for short) at each  ∈ ; (ii) if ∇ 2 () is positive definite (p.d. for short) at each  ∈ , then  is strictly convex.For convenience, we denote by Δ  the leading principal minors of .

Table 1 :
Information of interval of .