Exercise · ZW-OPTIMIZATION-PROBLEM-010 · ★★★ Advanced
Minimize \(f(x,y)=x^2+2y^2\) subject to \(x+y=3\). Give the minimum value and verify that it is global.
Choose one option.
Show answer
- D. 6
Explanation
With \(y=3-x\) we get \(x^2+2(3-x)^2\), whose minimum is at \(x=2\) and gives \(4+2=6\); since the function is convex, it is global. The value 9 corresponds to \(f(3,0)\), which is not the minimum.
Review the concept: Optimization problem
An optimization problem consists of minimizing or maximizing an objective function \(f\) over a set of points that satisfy certain constraints.
\[\min_{x \in \mathbb{R}^n} f(x) \quad \text{sujeto a} \quad g_i(x) \le 0\]
More practice problems: optimization problem
- ZW-OPTIMIZATION-PROBLEM-001 What is the general form of an optimization problem with inequality constraints?
- ZW-OPTIMIZATION-PROBLEM-002 What is the name of the function that is minimized in an optimization problem?
- ZW-OPTIMIZATION-PROBLEM-003 What is the name of the set of points that satisfy all the constraints?
- ZW-OPTIMIZATION-PROBLEM-004 What does it mean to minimize a function over a set?
- ZW-OPTIMIZATION-PROBLEM-005 What kind of problem is minimizing f(x)=x^2 with no constraints?
- ZW-OPTIMIZATION-PROBLEM-006 Minimize f(x,y)=x^2+y^2 subject to x+y=4. What is the minimum value?
- ZW-OPTIMIZATION-PROBLEM-007 Compute the minimum of f(x)=x^2-4x+7 over all reals.
- ZW-OPTIMIZATION-PROBLEM-008 Maximize f(x,y)=xy subject to x+y=10. What is the maximum value?