MATH 475 - Spring 2022 – Homework 5
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MATH 475 - Spring 2022 – Homework 5
Questions
1. Consider the optimization problem
minimize x2 + 1
subject to (x - 1)(x - 2) s 0.
(a) Find the minimal value px and the value of x that gives this minimum.
(b) Show that the Lagrangian dual function
h(a) = ,
a > -1
a s -1 .
(c) Show explicitly that strong duality holds for this problem.
2. Let 么 e 姓亿_n and 女 e 姓亿. Consider the optimization problem
minimize |≥|1
subject to 么≥ s 女
(here the inequality constraint means that (么≥); s b; for all i = 1, . . . , k).
(a) Show that the Lagrangian can be expressed as
L(≥, α) = |≥|1 + (Aoα) . ≥ - α . 女.
Deduce that
L(≥, α) > |≥|1 o1 - |么oα|* T - α . 女.
(b) Using part (a), find the Lagrangian dual function h(α). Hint: consider the two cases |么oα|* s 1 and |么oα|* > 1 separately.
(c) What is the dual problem?
2022-03-14