Stochastic Optimization Problems: Assignment Solutions and Methods

This assignment explores stochastic optimization problems, starting with a simple linear program with two variables and constraints. The solution analyzes scenarios with random variables, including cases where constraints hold with specified probabilities. The document then delves into Markowitz's ...Read More

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;
Min f(x,Ƹ)
Subject to: g(x,Ƹ)=0
h(x,Ƹ)≥ 0 ,

P(h(x,Ƹ)≥0)≥ p where p ∈[0,1] is the probability level.
You can design an optimization problem of the random constraints.
You can conclude the optimization model as;
Max ∑
1
2
ai 2 xi
Subject to:P(∑
1
2
aijxi≥ Ƹj (j=1……………….2)) ≥ p
Assumptions:
f is the objective function.
g is the inequality constraints,
x is the typical decision vector.
Ƹ is the vector of uncertainity.
Problem 3
Markowitz theory of mean-Variance optimization provides a technology of portfolio selection of
securities in a way that trades off the expected return and risk of of a potential portfolio.
Consider S1,,S2 with random return.
Assume that Ui is the expected return and σ I is the standard deviation of the assets Si and Sj.
Assume that u=[u1,,..u2]T
Expected return =E[X] =X1u1 +X2u2=uTx
Var[x]=∑
i= j
❑
pijσijxij =XT∑ x
Problem Formulation:
i)For Two Independent Random Coefficients:
Mean-Variance objective Function MaxxSin[u1TX +u2TX]

xT
∑
i
Xi ≤ σ2
Subject to : a11X1 + a12X2 =b1
a 21x1 +a22X2 =b2
ii)For Two Joint Random Coefficients:
Mean-Variance Objective Function Max Sin[uTX]
XT
∑ x =σ2
Subject to : a11X1 + a12X2 ≤b1
a 21x1 +a22X2 ≤b2
The objective function is risk -adjusted function is risk-aversion constant.
Problem 4
Non-linearities often arise when some of the coefficients in the model are random variables.
Suppose a Linear Programming having two constraints.
Maximize Sin (c1x2 +c2X2)
Subject to : a11X1 + a12X2 ≤b1
a 21x1 +a22X2 ≤b2
The coefficients a1 and a 2 are independently distributed .
Gi(y) represents that the random variable ai is at least as large as y.
The probability of both constraints being satisfied is at least β.
P[a11x1 +a22x2 ≤b1]x[a 21x1 +a22X2 ≤b2]≥ β
The condition can be written as the following sets of constraints.
-y1 + a110 + a120 ≤b1
-Y2 + a11 0+ a1202 ≤b1
G1(y1) X G2(y2)≥ β