Consider the problem:
P: min z=f(x)
gi(x) ≤ 0, i = 1,…,m
where x ∈ Rn, and f(x) is convex and the feasibility region is convex. The Sequential Unconstrained Minimization Technique is as follows:
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Step 1 Step 2
Disregarding the constraints of P, find a minimal point of the unconstrained problem. Is this point feasible? If yes, stop. Else, got to step 1.
Start with an initial interior feasible point x0 and set k = 1.
Use an unconstrained optimization method to find the minimum of the barrier function
B ( x , r ) = f ( x ) − r m 1
where, for instance, rk = 101−k. Start the minimization process from the point xk−1
and denote the the resulting optimal point by xk.
Does the current solution satisfy some stop criterion? If yes, stop. Else, set k := k + 1
and go to step 2.
Assuming tha the requirements for applying the above method are all satisfied, carry out 3 iterations to solve the problem:
P: min z = 3×21 −2x1x2 +2×2 −26×1 −8×2
x1+2×2 ≤6 x1−x2 ≥1 x1,x2 ≥ 0
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