Engineering Optimization: Theory and Practice, Fourth Edition

10.7 Sequential Linear Discrete Programming 615

Figure 10.7 Graphical solution of problem (E 6 ).

where the firstn 0 design variables are assumed to be discrete,dijis the jth discrete
value for the variablei, andX= {x 1 , x 2 ,... , xn}T. It is possible to find the solution
of this problem by solving a series of mixed-integer linear programming problems.
The nonlinear expressions in Eqs. (10.52) to (10.54) are linearized about a point
X^0 using a first-order Taylor’s series expansion and the problemis stated as

Minimizef (X)≈f (X^0 ) +∇f (X^0 )δX (10.57)

subject to

gj(X)≈gj(X^0 ) +∇gj(X^0 )δ X≤ 0 , j= 1 , 2 ,... , m (10.58)

hk(X)≈hk(X^0 ) +∇hk(X^0 )δ X= 0 , k= 1 , 2 ,... , p (10.59)

xi^0 + δxi∈ {di 1 , di 2 ,... , diq} , i= 1 , 2 ,... , n 0 (10.60)

xi(l)≤xi^0 +δxi≤x(u)i , i=n 0 + 1 ,n 0 + 2 ,... , n (10.61)

δX=X−X^0 (10.62)