Forcing a series of OR statements in ILP Problem

Question

I am attempting to solve an ILP program in relation to maximizing the return on investments. There are 10 decision variables, $x_1, x_2, ...,x_{10}$, with the following goals:

---

Max $0.067x_1 + ...+0.0590x_{10}$

Subject to:

• $x_1 + ... + x_{10} < 100,000$ (Budget)
• $x_1 \leq 25,000, ...,x_{10}\leq25,000$ (Divestment Constraint)
• $x_9 + x_{10} \geq x_1 + ...+x_4$
• $x_1+...+x_4 \leq 50,000$
• $x_1+...+x_4 \geq 20,000$

---

To fully capture the problem, I have another constraint I'd like to add: each decision variable has to be $0$, OR has to be at least $10,000$. Is there a good way through a system of inequalities to create this, or should I go backwards and make my decision variables binary to accomplish this?

Answer 1

I found a valid answer:

Introduce new decision variables $y_1,...,y_{10}$ with the constraints:

• $y_1,...,y_{10}$ are binary decision variables
• $x_i \geq 10,000*y_i, i = 1,2,...,10$
• Rewrite divestment constraint as $x_i \leq 25,000*y_i, i = 1,2,...,10$