Given the Following All-Integer Linear Program

question 37

Essay

Given the following all-integer linear program:
​
Max 15x₁ + 2x₂
s.t. 7x₁ + x₂ < 23
3x₁ - x₂ < 5
x₁,x₂ > 0 and integer
​
a.Solve the problem as an LP,ignoring the integer constraints.
b.What solution is obtained by rounding up fractions greater than or equal to 1/2? Is this the optimal integer solution?
c.What solution is obtained by rounding down all fractions? Is this the optimal integer solution? Explain.
d.Show that the optimal objective function value for the ILP (integer linear programming)is lower than that for the optimal LP.
e.Why is the optimal objective function value for the ILP problem always less than or equal to the corresponding LP's optimal objective function value? When would they be equal? Comment on the optimal objective function of the MILP (mixed-integer linear programming)compared to the corresponding LP and ILP.

---

Definitions:

Minimum Variance Portfolio

The minimum variance portfolio represents an investment portfolio constructed to achieve the lowest possible volatility or risk for its expected return, based on the correlation between assets.

Efficient

The ability to achieve a desired result without wasted energy or effort.

Optimal Portfolio

Optimal Portfolio is an investment portfolio that offers the highest expected return for a specific level of risk or the lowest risk for a given level of expected return.

Diversifiable Risk

A type of risk that can be reduced or mitigated through diversification or spreading investments across different assets to reduce exposure to any single risk.