Stochastic Optimization Techniques Report

Stochastic
Optimization
Problem 1
Consider a simple Linear Programming with 2 variables and 2 constraints.
Minimize Objective Function maxsin(C1X1 +C2X2)
Subject to :a11x1 + a12x2 ≤ b1 a=1,..2
a12x2 + a22x2 ≤ b2 b=1,..2
x1,x2 ≥0
But a11x1 + a12x2 ≤ b1 and a12x2 + a22x2 ≤ b2 hold a specified probability
of an unknown values of p1 and p2
Suppose alpha(α) =0.05
Minimize the objective function maxsin( C1X1 +C2X2)
Subject to :P1 +P2 >= 1-alpha
You can enumerate possibilities for x1=0 and x2=1,alpha =0.05
a 11 a 12 Is a110 + a211≤b1 ?
1 1 No P1
2 1 No p2
a 21 a 22 Is a21 + a22≤b2 ?
1 1 No P1
2 1 No P2
Hence x1 =0, X2 =0 is not a feasible solution.
Since the probability of ½ =0.5, you can suggest that the constraint is infeasible. It
is not possible for the constraint to have a feasible solution with the probability
0.95, as you can realize that 1-0.5 =0.5.
Thus the problem;

Minimize the objective function maxsin( C1X1 +C2X2)
Subject to ( a11x1 + a12x2 ≤ b1 and a12x2 + a22x2 ≤ b2 ) >=1-alpha
X,y>=0
Is now well-defined.
The problem contains only two variables which can easily be solved through
a simple search procedure. For instance, when α =0.05,the solution
x1=1,x2=2
When α= 0.01,x1=3, and x2=0
Problem 2
Consider b to be a random.
From a general formulation of an optimization problem under uncertainty of;