Exercise · ZW-OPTIMIZATION-PROBLEM-009 · ★★☆ Intermediate
Minimize \(f(x,y)=(x-1)^2+(y-2)^2\) subject to \(x+y\le 1\). What is the optimal point?
Choose one option.
Show answer
- A. \((0,1)\)
Explanation
The point \((1,2)\) is not feasible because \(1+2=3>1\). Projecting onto the line \(x+y=1\) gives \((0,1)\), at distance \(\sqrt{2}\) from \((1,2)\), less than the distance 2 of \((1,0)\).
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?