consider an optimization problem with 2 decision variables x

consider an optimization problem with 2 decision variables x1 and x2 the objective is to maximize z=c1x1 + 4x2
where c1 is a parameter. the feasible solution space is nonconvex and is shaded in the figure

a) determine all values of c1 fow which x1=2, x2=3 is an optimum solution

b) represent the feasible solution space by a set of linear constraints that should simultaneously be satisfied. add binary variables where needed.

Solution

Consider optimization problem   Maximize Z=c₁ x₁ +4x₂

An optimal solution to a lineaar program is a feasible solution with the largest objective value.

Non linear non convex optimization problems can have their optimal point everywhere.Non convex optimization may have multiple locally optimal points.

a) let x₁=2 and   x₂ =3 to determine the values of c₁ for an optimum solution

Z= 2c₁ +4(3) = therefore Z= 2c₁+12

c₁ can take up values such that Z should have maximum values therefore c₁ can have values greater than 3.

b) The linear constraints are

x₁ >-0 , x₂>-0

x₁ +x₂ >-4

x₁ +x₂ >-3