OPTIMIZATION 369
- Choose a suitable root finding method to determine the roots of tan x=x. How many roots of this equation
exist? Can all be determined?
- Study and suggest methods to determine multiple and/or complex roots of polynomial equations.
- Find and determine the nature (maximum or minimum) of the optimal point(s) of the function
fx( ) = x 12 + x 23 + x 32 subject to h(x) = x 1 + x 2 + x 3 – 1 = 0 using the Lagrangian multiplier method.
- Minimize fxxxxxx( ) = x 12 2 + 22 31 + 32 subject to hxxx( ) = + + – 6 = 0.x 123 Verify the sufficiency
condition.
- Minimize: xx 12 + 4 22
Subject to: x 1 – 2 ≥ 0
x 2 – 5 ≥ 0
Use KKT necessary conditions and sufficiency criterion to determine the nature of optimal solutions.
- Minimize: xx x 12 + 8 22 + 2 32
Subject to: x 1 + x 2 + x 3 – 15 ≤ 0
x 1 + x 2 + x 3 – 2 ≥ 0
Use sufficiency criterion to determine the nature of the optimal point(s).
- Solve Problem 8 using an additional constraint x 1 – x 2 + 2x 3 + 2 ≤ 0. Is the optimal solution (if it exists)
any different from that in Problem 8?
