Consider the following optimization problem.Minimizex∈IR2 x1 + x2Subject to 3x21 + x22 ≤ 3,x2 + 2x21 ≤ 0.(a) (5 points) Show that the MFCQ holds at every feasible point.(b) (30 points) Write down the KKT conditions and find all the stationary points
Question
Consider the following optimization problem.Minimizex∈IR2 x1 + x2Subject to 3x21 + x22 ≤ 3,x2 + 2x21 ≤ 0.(a) (5 points) Show that the MFCQ holds at every feasible point.(b) (30 points) Write down the KKT conditions and find all the stationary points
Solution
(a) The Mangasarian-Fromovitz Constraint Qualification (MFCQ) holds at a feasible point if the gradients of the active constraints at that point are linearly independent.
The constraints of the problem are:
- 3x1^2 + x2^2 ≤ 3
- x2 + 2x1^2 ≤ 0
The gradients of these constraints are:
- ∇(3x1^2 + x2^2) = [6x1, 2x2]
- ∇(x2 + 2x1^2) = [4x1, 1]
For the MFCQ to hold, these gradients must be linearly independent. This is true if the determinant of the matrix formed by these gradients is not zero. The determinant is:
6x11 - 2x24x1 = 6x1 - 8x1^2*x2
This is not zero for all feasible points (x1, x2) where x1 ≠ 0 and x2 ≠ 3/4x1. Therefore, the MFCQ holds at every feasible point.
(b) The Karush-Kuhn-Tucker (KKT) conditions are necessary conditions for a solution in nonlinear programming. They consist of the primal feasibility, dual feasibility, complementary slackness, and stationarity conditions.
The Lagrangian of the problem is:
L(x, λ) = x1 + x2 + λ1(3x1^2 + x2^2 - 3) + λ2(x2 + 2x1^2)
The KKT conditions are:
- Primal feasibility: The original constraints must hold.
- Dual feasibility: λ ≥ 0
- Complementary slackness: λi * gi(x) = 0 for all i
- Stationarity: The gradient of the Lagrangian with respect to x and λ must be zero.
The stationarity conditions give the following system of equations:
∂L/∂x1 = 1 + 6λ1x1 + 4λ2x1 = 0 ∂L/∂x2 = 1 + 2λ1x2 + λ2 = 0 ∂L/∂λ1 = 3x1^2 + x2^2 - 3 = 0 ∂L/∂λ2 = x2 + 2x1^2 = 0
Solving this system of equations will give the stationary points of the problem.
Similar Questions
Consider the following optimization problem:Minimizex∈IR3 2x21 + x22 + 2x23Subject to x1 + 2x2 + x3 ≤ −3.(a) (10 points) For each c > 0, defineqc(x) := 2x21 + x22 + 2x23 + c2 (x1 + 2x2 + x3 + 3)2+Argue that qc is convex and find the global minimizer of qc.(b) (10 points) For each µ > 0, defineℓµ(x) := 2x21 + x22 + 2x23 − µ ln (−3 − x1 − 2x2 − x3) .Argue that ℓµ is convex and find the global minimizer of ℓµ. You may use without proof thefact that the function t 7 → − ln(−3 − t) is convex (as an extended real-valued function).
Optimality. Consider the following optimization problemminimize 12 xT P x + qT x + rsubject to −1 ≤ xi ≤ 1, i = 1, 2, 3whereP = 26 24 −424 34 12−4 12 24 , q =−44−2926 , r = 1.(a) Prove that x? = (1, 0.5, −1) is optimal.(b) Are any of the constraints inactive at x?? If so, which one(s)?
Consider the optimisation problemminx∈R3x1−2x2+ex3subject toAx=0min𝑥∈𝑅3𝑥1−2𝑥2+𝑒𝑥3subject to𝐴𝑥=0where A𝐴 is a 2×32×3 matrix with rank two. Moreover, the sum of the entries in each row of A𝐴 is zero.(a) This problem can be rewritten as an unconstrained convex optimisation problem in how many variables? Answer 1 Question 1(b) The optimal solution to the original problem, in the variables x1,x2𝑥1,𝑥2 and x3𝑥3, is given by:
Consider the following constrained optimization problem:max f (x, y) = x2 + x + 4y2subject to 2x + 2y ≤ 1x ≥ 0y ≥ 0.a. Suppose we have already checked NDCQ holds for this problem. Write the originalLagrangian function.b. Find the solution to the problem by using the Lagrangian function written above.11c. Write the Kuhn-Tucker Lagrangian function and the associated (first-order) equationsor conditions. (No need to solve them.)12d. Name at least one advantage of using the Kuhn-Tucker formulation.13e. Approximate the new max value if we change the objective function to x2 + 1.2x + 4y2.f. Approximate the new max value if we change the objective function to x2 + x + 4.2y2.4
Consider the optimisation problem minx∈R3x1−2x2+ex3subject toAx=0 where A is a 2×3 matrix with rank two. Moreover, the sum of the entries in each row of A is zero. (a) This problem can be rewritten as an unconstrained convex optimisation problem in how many variables? Answer 1 Question 1 0 (b) The optimal solution to the original problem, in the variables x1,x2 and x3 , is given by:
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.