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Engineering Optimization
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Global optimization for signomial
discrete programming problems in
engineering design
Jung-Fa Tsai a , Han-Lin Li a & Nian-Ze Hu a
a
Institute of Information Management , National Chiao Tung University , Taiwan, Republic of China
Published online: 17 Sep 2010.
To cite this article: Jung-Fa Tsai , Han-Lin Li & Nian-Ze Hu (2002) Global optimization for signomial discrete programming problems in engineering design, Engineering Optimization, 34:6, 613-622 To link to this article: http://dx.doi.org/10.1080/03052150215719
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Eng. Opt., 2002, Vol. 34, pp. 613–622
GLOBAL OPTIMIZATION FOR SIGNOMIAL
DISCRETE PROGRAMMING PROBLEMS
IN ENGINEERING DESIGN
JUNG-FA TSAI, HAN-LIN LI* and NIAN-ZE HUInstitute of Information Management, National Chiao Tung University, Taiwan, Republic of China
(Received 26 November 2001; In final form 8 May 2002)
This paper proposes a novel method to solve signomial discrete programming (SDP) problems frequently occurring in engineering design. Various signomial terms are first convexified following different strategies. The original SDP program is then converted into a convex integer program solvable by commercialized packages to obtain globally optimal solutions. Compared with current SDP methods, the proposed method is guaranteed to converge to a global optimum, is computationally more efficient, and is capable of treating zero boundary problems. Numerical examples are presented to demonstrate the usefulness of the proposed method in engineering design.
Keywords: Signomial discrete programming problem; Global optimization; Convexification
1 INTRODUCTION
Signomial discrete programming (SDP) problems occur quite frequently in various fields such as civil and material engineering design, chemical engineering, location-allocation, inventory control, production planning, and scheduling etc. These applications are extensively reviewed in Floudas and Pardalos [9] and Floudas [6]. The developed methods for SDP can be divided into three approaches. The first SDP approach includes various heuristic techniques. For instance, Salcedo et al. [18] propose an improved random search algorithm for solving nonlinear optimiza-tion problems. Cardoso et al. [2] solve nonconvex nonlinear integer programming problems with simulated annealing. Wang and Liao [21] develop methods for solving polynomial integer pro-grams by the genetic algorithm. Their methods, however, can only guarantee to find local optima. Moreover, the probability of finding the global solution decreases when the problem size increases. The second SDP approach for global optimization is the use of stochastic methods such as the Multi-Level Single Linkage method proposed by Rinnooy and Timmer [17] and the Multistart method proposed by Li and Chou [12]. These techniques have a high probability of finding a global optimum for a SDP problem. However, since this approach requires the eva-luation of a large number of starting points, it can only be applied to solve small size problems. The third approach is the deterministic method. Duran and Grossmann [4] treat a class of SDP problems by outer approximation techniques. Michelon and Maculan [15] solve SDP problems by
* Corresponding author. E-mail: hlli@cc.nctu.edu.tw
ISSN 0305-215X print; ISSN 1029-0273 online # 2002 Taylor & Francis Ltd DOI: 10.1080=0305215021000063237
Lagrangean decomposition techniques. Li and Chang [11] solve SDP problems, where all signomial terms have integer power values, by piecewise linearization techniques. Po¨rn et al. [16] introduce different convexification strategies for SDP problems with both posynomial and nega-tive binomial terms in the constraints. The above methods, however, can only handle some specially-structured SDP problems. Recently, Floudas and Pardalos [9], Maranas and Floudas [14], and Floudas ½7; 8 have proposed more general methods to treat SDP problems. Their meth-ods have been applied widely to solve engineering design problems. The core concept of Floudas’s approach is to convert a SDP problem into a new problem in which both the constraints and the objective are decomposed into the difference of two convex functions. By utilizing expo-nential variable transformation, Floudas’s method transform each signomial term z ¼ xa
1x b 2, where
x1and x2are positive integers, into an exponential term z0¼ea ln x1þb ln x2. Since (i) the
exponen-tiation of a linear expression is convex, and (ii) ln x1and ln x2can be conveniently expressed using
0–1 variables, the signomial term can then be fully expressed as the combination of convex integer terms. Floudas’s method therefore can find the global optimum of a SDP problem successfully. However, since Floudas’s method performs exponential transformation for all product terms, it requires the use of a large number of 0–1 variables to piecewisely linearize the logarithmic terms. In addition, the exponential transformation technique can only be applied to positive vari-ables and is unable to treat zero boundary problems where varivari-ables might have zero value.
This paper proposes another method to treat SDP problems and develops several strategies for convexifying a signomial term. The advantages of the proposed methods over the current SDP methods mentioned above are given below:
(i) Compared with the heuristic approaches and the stochastic methods of Duran and Grossmann [4], and Michelon and Maculan [15], the proposed method is guaranteed to find a global optimum of a SDP problem.
(ii) Compared with Floudas’s method, for many cases, the proposed method uses fewer extra 0–1 variables to linearize a signomial term. In addition, the proposed method can treat non-negative integer variables while Floudas’s method can only treat positive integer variables. This study first discusses some theoretical propositions about SDP programs. The rules of convexification are then proposed. Following that, some numerical examples of engineering design are solved to demonstrate the usefulness of the proposed method.
2 THEORETICAL DEVELOPMENT
A Signomial Discrete Programming (SDP) problem discussed here is formulated below: P1 Minimize ZðX Þ ¼X P cpzp Subject to X q hkqzkqlk; k ¼ 1; 2; . . . ; K Ym j¼1 fjðxÞ ¼ 0; j ¼ 1; 2; . . . ; m zp¼xapp11x ap2 p2 x apmðpÞ pmðpÞ zkq¼x bkq1 kq1x bkq2 kq2 x bkqmðkqÞ kqmðkqÞ X ¼ ðx1; x2;. . . ; xnÞ; 0 xixi xxi; xi2X
are non-negative discrete variables.
In problem P1, cp, api, bkqi, hkq, lk are constants and unrestricted in sign, xi and xxi are
respectively the lower and upper bounds of discrete variables xi.
P1 is a nonconvex integer problem which can only be solved to obtain the local optimum. In order to obtain its global optimum, P1 must be converted into a convex integer problem. The conventional convex integer techniques proposed by Borchers and Mitchell [1] and Floudas [5] have traditionally been used to solve convex integer programs to obtain global optima. This paper proposes various techniques for convexifying signomial terms zp, zp,
zkq, and zkq. The convexified SDP program can be expressed as a linear integer
program-ming problem solvable by many commercialized optimization packages to obtain a globally optimal solution. Some propositions related to convexification techniques are described as follows.
PROPOSITION1 For positive discrete variables xi2 fdi1; di2;. . . ; dinigwhere di; jþ1> dij> 0 for j ¼ 1; 2; . . . ; ni1; a product term xr11x
r2
2 xrnn with r1; r2;. . . ; rnreal constants can be
transformed to a function er1y1þþrnyn where y
i¼ln di1þ
Pni1
j¼1 uijðln di; jþ1ln di1Þ;
Pni1
j¼1 uij1 for uij2 f0; 1g.
Proof Let xi¼eyi and yi¼ln xi, expressing xias
xi¼di1þ X ni1 j¼1 uijðdi; jþ1di1Þ; X ni1 j¼1 uij1; where uij2 f0; 1g: We then have xr1 1x r2 2 xrnn ¼er1y1þþrnyn and yi¼ln di1þPj¼1ni1uijðln di; jþ1ln di1Þ, Pni1 j¼1 uij1, for uij2 f0; 1g.
Suppose a variable xiin Proposition 1 may have zero value, then Proposition 1 needs to be
modified as in the following proposition:
PROPOSITION 2 For non-negative discrete variables xi2 f0; di1; di2;. . . ; dinig where di; jþ1 > dij> 0 for j ¼ 1; 2; . . . ; ni1; then a product term z ¼ xr11x
r2 2 xrnn can be expressed as (i) 0 z zzðPni j¼1uijÞ, (ii) zzðPni¼1Pni j¼1uijnÞ þ er1y1þþrnynz zzðn Pni¼1 Pni j¼1uijÞ þLðer1y1þþrnynÞ, where xi¼ Pni j¼1uijdij, yi¼ Pni j¼1uijðln dijÞ, Pni j¼1uij1, uij2 f0; 1g, Lðer1y1þþrnynÞ is a
piecewisely linearized expression of er1y1þþrnyn, and zz is the upper bound of z. Proof If there is xi¼0 for some i, then
Pni
j¼1uij¼0 and z ¼ 0. If xi> 0 for all
i ¼ 1; 2; . . . ; n, thenPni¼1Pni
j¼1uij¼n and er1y1þþrnyn z Lðer1y1þþrnynÞ. Therefore, the
proposition is then proven.
Remark 1 For a discrete variable x, x 2 fd1; d2;. . . ; dng, d1; d2;. . . ; dn are positive values,
the exponential term xa where a is a real constant can be represented as
xa¼d1aþX n1 j¼1 ujðdjþ1a d a 1Þ where Xn1 j¼1 uj1; uj2 f0; 1g:
SIGNOMIAL DISCRETE OPTIMIZATION 615
PROPOSITION3 A product term z ¼ uf ðxÞ is equivalent to the following linear inequalities
(i) M ðu 1Þ þ f ðxÞ z M ð1 uÞ þ f ðxÞ, (ii) Mu z Mu,
u 2 f0; 1g, z is an unrestricted in sign variable, and M ¼ max f ðxÞ is a large constant. Proof If u ¼ 1 then z ¼ f ðxÞ, and if u ¼ 0 then z ¼ 0.
Remark 2 The product term u1u2 um where ui2 f0; 1g for i ¼ 1; 2; . . . ; m can be
replaced by a variable u expressed as (i) 0 u ui, for i ¼ 1; 2; . . . ; m,
(ii) u Pmi¼1uim þ 1.
Proof If ui¼0 for any i, then u ¼ 0. If ui¼1 for all i, then u ¼ 1.
PROPOSITION 4 A twice-differentiable function f ðx1; x2; x3Þ ¼ xa1x b 2x
g
3 is convex for
a þ b þ g 1 where x1; x2; x3;a; b; g 0.
Proof Denote Hðx1; x2; x3Þas the Hessian matrix of f ðx1; x2; x3Þ.
Hðx1; x2; x3Þ ¼ q2f ðx1; x2; x3Þ qx1qx1 q2f ðx1; x2; x3Þ qx1qx2 q2f ðx1; x2; x3Þ qx1qx3 q2f ðx1; x2; x3Þ qx2qx1 q2f ðx1; x2; x3Þ qx2qx2 q2f ðx1; x2; x3Þ qx2qx3 q2f ðx1; x2; x3Þ qx3qx1 q2f ðx1; x2; x3Þ qx3qx2 q2f ðx1; x2; x3Þ qx3qx3 2 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 5 ¼ aða 1Þxa2 1 x b 2x g 3 abxa11 x b1 2 x g 3 agxa11 x b 2x g1 3 abxa1 1 x b1 2 x g 3 bðb 1Þxa1x b2 2 x g 3 bgxa1x b1 2 x g1 3 agxa1 1 x b 2x g1 3 gbxa1x b1 2 x g1 3 gðg 1Þxa1x b 2x g2 3 2 6 4 3 7 5
The ith principal minor, denoted by Hi, of a n n matrix is the i i matrix obtained by
deleting the last n i rows and columns of the matrix. It is clear that if det H10,
det H20, and det H30, then f ðx1; x2; x3Þis convex.
Check: (i) det H10 (
:_:
x1; x2; x3;a; b; g 0 and aða 1Þxa21 x b 2x g 30). (ii) det H20 ( :_:
det H2¼abx2a21 x 2b2 2 x 2g 3 ða b þ 1Þ 0). (iii) det H3 0 ( :_:
det H3¼abgx3a21 x 3b2
2 x
3g2
3 ða b g þ 1Þ 0).
Following (i), (ii), and (iii), the proposition is proven.
PROPOSITION 5 An equality constraint Qmj¼1fjðxÞ ¼ 0 can be replaced by following
expressions.
(i) M ð1 ujÞ< fjðxÞ < M ð1 ujÞ,
(ii) Pmj¼1uj1,
where M is a large constant, M ¼ max f0; fjðxÞg, and uj2 f0; 1g for j 2 f1; 2; . . . ; mg.
Proof Expression (i) means if and only if uj¼1 then fjðxÞ ¼ 0. Expression (ii) means there
is at least one j 2 f1; 2; . . . ; mg such that uj¼1. Both expressions ensureQmj¼1fjðxÞ ¼ 0.
3 CONVEXIFICATION STRATEGIES
Following the above discussion, a signomial term with three variables is here used as an example to describe the strategy of convexification. The strategy can be extended to convexify a signomial term containing n variables.
Consider a signomial term cxa1xb2xg3composed of three positive discrete variables x1; x2; x3,
where xi¼di1þPnj¼1i1uijðdi; jþ1di1Þ, Pnj¼1i1uij1. This term can be convexified by
following rules:
Rule 1 If c > 0, then let cxa1xb2xg3¼cea ln x1þb ln x2þg ln x3 where ln x
i¼ln di1þ
Pni1
j¼1 uijðln di; jþ1ln di1Þ,
Pni1
j¼1 uij1, for uij2 f0; 1g.
Rule 2 If c < 0, a; b; g 0, and a þ b þ g 1, then cxa 1x
b 2x
g
3 is already a convex term
following Proposition 4. No convexification is required.
Rule 3 If c < 0, 0 a; b < 1, g 0, a þ b < 1, and a þ b þ g > 1, then let cxa 1x
b 2x
g 3¼
cxa1xb2y1ab3 and y3¼xg=ð1abÞ3 where cxa1x b 2y
1ab
3 is regarded as a convex term, and
y3 is a discrete variable, y3¼h31þPnj¼131u3jðh3;jþ1h31Þ, h3j¼ ðd3jÞg=ðaþbþgÞ for
j 2 f1; 2; . . . ; ni1g.
Rule 4 If c < 0, a; b; g > 0, and a þ b þ g > 1, then let cxa1xb2x3g¼cya=ðaþbþgÞ1 yb=ðaþbþgÞ2 yg=ðaþbþgÞ3 where y1¼xaþbþg1 , y2¼xaþbþg2 , y3¼xaþbþg3 . cy
a=ðaþbþgÞ 1 y b=ðaþbþgÞ 2 y g=ðaþbþgÞ 3 is a
convex term, and yi¼hi1þPnj¼1i1uijðhi; jþ1hi1Þ, hij¼ ðdijÞg=ðaþbþgÞ for i ¼ 1; 2; 3, and
j 2 f1; 2; . . . ; ni1g.
Rule 5 If a; b > 0, x3 ¼1, and a þ b > 1, then let cxa1x b 2 ¼c½d11a þ Pn11 j¼1 u1jðd1; jþ1a da 11Þx b
2 for j 2 f1; 2; . . . ; n11g. By Proposition 3, the product term u1jxb2 can be
trans-formed into linear inequalities.
4 NUMERICAL EXAMPLES
According to the convexification strategies described above, several examples are presented in the following to demonstrate its usefulness in engineering design.
Example 1 Consider the following nonconvex minimization problem containing three integer variables.
Minimize x21x3:52 x3x2x2:63 x31
Subject to x1þx2þx310
1 x15; 1 x25; 1 x35; x1; x2; x3 are inter variables:
This program is a nonconvex integer program. Solving it by LINGO 7.0 [13], the obtained solution is ðx1; x2; x3Þ ¼ ð1; 2; 5Þ and the objective value is 75:7579. This is, however, a
SIGNOMIAL DISCRETE OPTIMIZATION 617
local optimum. In order to obtain a global optimum, all signomial terms are transformed into convex terms as follows:
(i) x2
1x3:52 x3 is convexified as e2y1þ3:5y2þy3 by Rule 1.
(ii) x2x2:63 is convexified as
x2x2:63 ¼ ð1 þ u21þ2u22þ3u23þ4u24Þx2:63
¼ h1z12z23z34z4 by Rule 5:
(iii) x31 is treated directly as
x31¼ 1 ð231Þu11 ð331Þu12 ð431Þu13 ð531Þu14
¼ 1 7u1126u1263u13124u14:
The transformed program is then presented as a convex integer program below: Minimize e2y1þ3:5y2þy3h
1z12z23z34z4h2 Subject to x1þx2þx310 x1¼1 þ u11þ2u12þ3u13þ4u14 y1¼u11ln 2 þ u12ln 3 þ u13ln 4 þ u14ln 5 u11þu12þu13þu141 x2¼1 þ u21þ2u22þ3u23þ4u24 y2¼u21ln 2 þ u22ln 3 þ u23ln 4 þ u24ln 5 u21þu22þu23þu241 x3¼1 þ u31þ2u32þ3u33þ4u34 y3¼u31ln 2 þ u32ln 3 þ u33ln 4 þ u34ln 5 u31þu32þu33þu341
h1¼1 þ ð22:61Þu31þ ð32:61Þu32þ ð42:61Þu33þ ð52:61Þu34
h2¼1 þ ð231Þu11þ ð331Þu12þ ð431Þu13þ ð531Þu14
M ðu211Þ þ h1 z1M ð1 u21Þ þh1 0 z1Mu21
M ðu221Þ þ h1 z2M ð1 u22Þ þh1 0 z2Mu22
M ðu231Þ þ h1 z3M ð1 u23Þ þh1 0 z3Mu23
M ðu241Þ þ h1 z4M ð1 u24Þ þh1 0 z4Mu24
ð1; 1; 1; 0; 0; 0Þ ðx1; x2; x3; y1; y2; y3Þ ð5; 5; 5; ln 5; ln 5; ln 5Þ
where uij2 f0; 1g; M is a large constant:
Solving the above convex integer program by LINGO 7.0 [13], the obtained global optimal solution is ðx1; x2; x3Þ ¼ ð5; 1; 1Þ and the objective value is 101.
If we let 0 x15, 0 x25, 0 x35, Example 1 becomes a nonconvex integer
problem with non-negative variables. Floudas’s method, however, cannot be used to solve this kind of problem. By Proposition 2, we can treat zero boundary problems effectively. Solving a modified Example 1 with non-negative variables by LINGO 7.0 [13] yields the global solution ðx1; x2; x3Þ ¼ ð5; 4; 0Þ and the objective value is 125.
Example 2 Consider the optimal design problem of a pressure vessel given in Sandgren [19] depicted in Figure 1 where x1(the spherical head thickness) and x2(the shell thickness)
are discrete variables and x3 (the radius of the shell) and x4 (the length of the shell) are
continuous variables. This problem was solved by Sandgren [19] and Fu et al. [10] to obtain a locally optimal solution. Li and Chou [12] and Li and Chang [11] solved this problem to obtain an approximate solution. In order to illustrate the applicability of the present method in solving signomial discrete programs, all variables x1, x2, x3, and x4are treated as discrete
variables. The problem is formulated below:
Minimize 0:6224x1x3x4þ1:7781x2x32þ3:1661x21x4þ19:84x21x3 Subject to x1þ0:0193x3 0 x2þ0:00954x30 px2 3x4 4 3px 3 3þ750 1728 0 240 þ x40 1 x11:375 0:625 x21 48 x352 90 x4112
where x1 and x2 are discrete variables with discreteness 0.0625, and x3 and x4 are integer
variables.
x1 is the spherical head thickness, x2 is the shell thickness, x3 is the radius and x4 is the
length of the shell. The product term x1x3x4 can be treated by Rule 1; product terms x2x23,
x21x4, and x21x3 can be treated by Rule 5. x1; x2; x3 and x4 can be completely expressed by
binary variables as follows:
x1 ¼1 þ 0:0625u11þ0:125u12þ0:25u13
x2 ¼0:625 þ 0:0625u21þ0:125u22þ0:25u23
x3 ¼48 þ u31þ2u32þ4u33
x4 ¼90 þ u41þ2u42þ4u43þ8u44þ16u45; uij2 f0; 1g
This program can then be completely transformed to a convex 0–1 program solvable to obtain a globally optimal solution. A detail description about how to solve a convex 0–1
FIGURE 1 Tube and pressure vessel (Sandgren [19]).
SIGNOMIAL DISCRETE OPTIMIZATION 619
program can be found in Borchers and Mitchell [1] and Floudas [5]. By utilizing a branch-bound algorithm or an outer approximation algorithm, a convex 0–1 program can be solved conveniently to reach a global optimum. A commercialized optimization package (i.e. LINGO [13]) is also available for solving convex integer programs. Solving this program by LINGO 7.0 [13], the obtained global solution is ðx1; x2; x3; x4Þ ¼ ð1; 0:625; 51; 91Þ and
the objective is 7079.037. The comparison of the solutions for this example is given in Table I. Table I illustrates that even with the extra restriction of the discreteness requirements on the variables x3and x4, the present method obtains a better solution than other methods do.
Example 3 This example shows the detailed process of solving a global nonlinear mixed discrete programming (GDP) problem with Proposition 5. The problem is modified from Cha and Mayne [3]. Minimize 2x2 1þx3216x1x210x2 Subject to ðx216x1þ4x211Þ½ð3:25x13:1x2Þ2þ ðx1þx26:35Þ2 ½ð3:55x13:3x2Þ2þ ðx1þx26:85Þ2½ð3:6x13:5x2Þ2þ ðx1þx27:1Þ2½ð3:8x14:1x2Þ2þ ðx1þx27:9Þ22¼0 x1x2þ3x2þex131 0 3 x16 3 x25
where x1 is an integer variable and x2 is a discrete variable with discreteness 0.2.
x1 and x2 are expressed as:
x1 ¼3 þ u11þ2u12; u11; u122 f0; 1g
x2 ¼3 þ 0:2u21þ0:4u22þ0:8u23þ1:6u24; u21; u22; u23; u24 2 f0; 1g
Here the product term x1x2 can be treated by Rule 5, and the first constraint can be
trea-ted by Proposition 5. This program can then be convertrea-ted into a linear integer program. By Proposition 5, the first constraint in the program can be reformulated with following inequa-lity constraints. M ð1 u1Þ x216x1þ4x211 M ð1 u1Þ M ð1 u2Þ ð3:25x13:1x2Þ2þ ðx1þx26:35Þ2 M ð1 u2Þ M ð1 u3Þ ð3:55x13:3x2Þ2þ ðx1þx26:85Þ2 M ð1 u3Þ M ð1 u4Þ ð3:6x13:5x2Þ2þ ðx1þx27:1Þ2 M ð1 u4Þ M ð1 u5Þ ½ð3:8x14:1x2Þ2þ ðx1þx27:9Þ22M ð1 u5Þ
u1þu2þu3þu4þu51; where M is a large constant, uj2 f0; 1g; j ¼ 1; 2; . . . ; 5.
TABLE I A Comparison of Optimum Solutions for Example 2.
Items Sandgren Fu et al.
Li and Chou Li and Chang The proposed method x1 1.125 1.125 1 1 1 x2 0.625 0.625 0.625 0.625 0.625 x3 48.95 48.38 51.25 51.25 51 x4 106.72 111.745 90.991 90.991 91 Objective 7982.5 8048.6 7127.3 7127.3 7079.037
Solving the transformed program with LINGO 7.0 [13], the global optimal solution is found as ðx1; x2Þ ¼ ð5; 4Þ and the objective value is 246.
Example 4 This example is an optimal design problem introduced in Shin et al. [20]. This is a three-bar truss design problem as depicted in Figure 2. The indeterminate three bar truss is subject to vertical and horizontal forces. The weight is to be minimized under the constraint that the stress in all members should be smaller than the allowable stress. The problem can be stated as follows. Minimize 2x1þx2þ ffiffiffi 2 p x3 Subject to 1 þ ffiffiffi 3 p x2þ1:932x3 1:5x1x2þ ffiffiffi 2 p x2x3þ1:319x1x3 0 1 þ 0:634x1þ2:828x3 1:5x1x2þ ffiffiffi 2 p x2x3þ1:319x1x3 0 1 þ 0:5x12x2 1:5x1x2þ ffiffiffi 2 p x2x3þ1:319x1x3 0 1 0:5x12x2 1:5x1x2þ ffiffiffi 2 p x2x3þ1:319x1x3 0
where xi are discrete variables, xi2 f0:1; 0:2; 0:3; 0:5; 0:8; 1:0; 1:2g; i ¼ 1; 2; 3.
This problem is nonconvex because of the constraints. The nonconvex terms x1x2,
x1x3, and x2x3 can be treated by Rule 4. The problem is then transformed into an
equiva-lent convex integer program as follows. Minimize 2x1þx2þ ffiffiffi 2 p x3 Subject to pffiffiffi3x2þ1:932x31:5X10:5X20:5 ffiffiffi 2 p X0:5 2 X30:51:319X10:5X30:50 0:634x1þ2:828x31:5X10:5X20:5 ffiffiffi 2 p X0:5 2 X30:51:319X10:5X30:50 0:5x12x21:5X10:5X20:5 ffiffiffi 2 p X20:5X30:51:319X10:5X30:50 0:5x1þ2x21:5X10:5X20:5 ffiffiffi 2 p X0:5 2 X30:51:319X10:5X30:50
x1¼0:1 þ 0:1u11þ0:2u12þ0:4u13þ0:7u14þ0:9u15þ1:1u16
x2¼0:1 þ 0:1u21þ0:2u22þ0:4u23þ0:7u24þ0:9u25þ1:1u26
x3¼0:1 þ 0:1u31þ0:2u32þ0:4u33þ0:7u34þ0:9u35þ1:1u36
X1¼0:01 þ 0:03u11þ0:08u12þ0:24u13þ0:63u14þ0:99u15þ1:43u16
X2¼0:01 þ 0:03u21þ0:08u22þ0:24u23þ0:63u24þ0:99u25þ1:43u26
X3¼0:01 þ 0:03u31þ0:08u32þ0:24u33þ0:63u34þ0:99u35þ1:43u36
u11þu12þu13þu14þu15þu16 1
u21þu22þu23þu24þu25þu26 1
u31þu32þu33þu34þu35þu36 1
FIGURE 2 Three bar truss for Example 4 (Shin et al. [20]).
SIGNOMIAL DISCRETE OPTIMIZATION 621
where uij2 f0; 1g, xi are discrete variables, xi2 f0:1; 0:2; 0:3; 0:5; 0:8; 1:0; 1:2g; i ¼ 1; 2; 3,
and j ¼ 1; 2; . . . ; 6.
Solving this convex integer program by LINGO 7.0 [13] gives the global optimal solution ðx1; x2; x3Þ ¼ ð1:2; 0:5; 0:1Þ and the objective value 3.0414. Shin et al. [20] and Li and
Chou [12] solved this problem and got the same solution. Their methods, however, cannot claim the solution found is a global optimum.
5 CONCLUSIONS
This study proposes global optimization techniques to obtain the global optimal solutions of several types of SDP problems. Different convexification techniques for SDP problems were presented. The transformation methods are general and practical for many kinds of noncon-vex global optimization problems. The numerical examples chosen from the literature demonstrate that the proposed methods can obtain the global solutions effectively.
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