In addition, nonlinear SHE equations resemble nonlinear constrained optimization problems having objective function and
constraint function [1], [2].
To enforce our prior knowledge of the structure of y, we specify a weighted
constraint function g, which penalizes output structures that are not consistent with our understanding of the task.
where [C.sub.slab] is the
constraint function for the slab, Mst is the minimum adequate thickness for the slab, and St is the slab thickness.
In quadratic programming for maximizing with interval variables, the best optimum problem has properties as the best version of the objective function and the maximum feasible area on the
constraint function, whereas the worst optimum problem has properties as the worst version of the objective function and the minimum feasible area on the
constraint function.
And gi(x)[less than or equal to]0 is the
constraint function, p is the number of constrains.
[[lambda].sub.s] denotes a scaling factor that determines the minimum step length for the [S.sub.DV] for each evaluation of the limit-state or
constraint function m is a maximum integer or decimal value that is needed to be found in the optimization loop because the m might help to search for a minimum [[mu].sup.r.sub.DV].
Further, Duffin and Peterson (1972) pointed out that each of those posynomial programs GP can be reformulated so that every
constraint function becomes posy-/bi-nomial, including at most two posynomial terms, where posynomial programming--with posy-/mo-nomial objective and constraint functions--is synonymous with linear programming.
Optimization results Initial Case 1 W, H(x) Case 2 W, H(y) CPU time 18h40 18h 16 Iterations 0 3 3 Objective function f 1 0.134 0.14 Improvement of the velocity distribution [%] - 87 86
Constraint function P/PO 1 0.92 0.97 Global relative deviation E [%] 19.77 2.65 2.77 Global relative deviation of the average velocities [E.sub.R.sup.s] 115.25 14.68 13.2 Variable W [mm] 20 8.03 8 Variable H [mm] 7 10.36 7.23 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
In this paper, we propose an approximate algorithm based on membrane computing for constrained optimization problems: a membrane associates a constraint and the tentative solutions evolve according to the rules in the membrane and are evaluated by the
constraint function value as the fitness; the sub-populations can communicate efficiently during the evolution process by making use of the structure of P systems and the communication mechanism among the membranes.
Constrained optimization: The color essentially codes the fourth variable related to the perimeter constraint--the contours of the
constraint function are represented by colors.