Reference no: EM132300000

Assignment - Numerical Optimization Problems

Textbook - Numerical Optimization, Second Edition by Jorge Nocedal and Stephen J. Wright. ISBN-10: 0-387-30303-0.

Problem 1 - do problem 16.17: Consider the quadratic program

max 6x₁ + 4x₂ - 13 - x₁² - x₂²,

subject to x₁ + x₂ ≤ 3, x₁ ≥ 0, x₂ ≥ 0.

First solve it graphically, and then use your program implementing the active-set method given in Algorithm 16.3. (Algorithm 16.3 Active-Set Method for Convex QP).

Problem 2 - do problem 17.8: Prove the second part of Theorem 17.4. That is, if xˆ is a stationary point of φ₁(x; µ) for all µ sufficiently large, but xˆ is infeasible for problem (17.6), then xˆ is an infeasible stationary point. (Hint: Use the fact that D(φ₁(xˆ; µ); p) = ∇ f (xˆ)^T p + µD(h(xˆ); p), where h is defined in (17.27).)

Problem 3 - do problem 19.7:  Program the simple interior-point method Algorithm 19.1 and apply it to the problem (18.69). Use the same starting point as in that problem. Try different values for the parameter σ. (Algorithm 19.1 - Basic Interior-Point Algorithm).