Prof Dr Jörg Fliege Semester 2, 2022/2023
School of Mathematical Sciences
University of Southampton
MATH6184 - Optimization Part
Exercise Sheet 1
You are not required to hand in the answers to these problems.
1. Consider the problem
min 2(x − 3)2
subject to 0 ≤ x ≤ 6.
(a) Verify that this problem is a convex problem.
(b) Verify that the unique solution occurs at x = 3.
(c) Verify that x = 0 and x = 6 are alternate optima if the objective is maximised
instead of minimized.
2. Determine wether each of these nonlinear programming problems is a convex program:
(a)
max ln x1 + 3x2 − x32
subject to x1 ≥ 0,
2x1 + 3x2 − x3 = 1,
x21 + x22 ≤ 9.
(b)
max x1 + x2
subject to x1 x2 ≤ 9,
−5 ≤ x1 ≤ 5,
−5 ≤ x2 ≤ 5.
(c)
max x1 + 6/x1 + 5x22
subject to 4x1 + 6x2 ≤ 35,
x1 ≥ 5,
x2 ≥ 0.
3. A lidless, rectangular box is to be manufactured from cardboard. The cardboard is 30cm
wide and 40cm long.The box is manufactured by cutting away squares from the four
corners of the cardboard, folding up ends and sides, and joining them with heavy type.
The designer wishes to choose box dimensions that maximise the volume of the box.
1
School of Mathematical Sciences
University of Southampton
MATH6184 - Optimization Part
Exercise Sheet 1
You are not required to hand in the answers to these problems.
1. Consider the problem
min 2(x − 3)2
subject to 0 ≤ x ≤ 6.
(a) Verify that this problem is a convex problem.
(b) Verify that the unique solution occurs at x = 3.
(c) Verify that x = 0 and x = 6 are alternate optima if the objective is maximised
instead of minimized.
2. Determine wether each of these nonlinear programming problems is a convex program:
(a)
max ln x1 + 3x2 − x32
subject to x1 ≥ 0,
2x1 + 3x2 − x3 = 1,
x21 + x22 ≤ 9.
(b)
max x1 + x2
subject to x1 x2 ≤ 9,
−5 ≤ x1 ≤ 5,
−5 ≤ x2 ≤ 5.
(c)
max x1 + 6/x1 + 5x22
subject to 4x1 + 6x2 ≤ 35,
x1 ≥ 5,
x2 ≥ 0.
3. A lidless, rectangular box is to be manufactured from cardboard. The cardboard is 30cm
wide and 40cm long.The box is manufactured by cutting away squares from the four
corners of the cardboard, folding up ends and sides, and joining them with heavy type.
The designer wishes to choose box dimensions that maximise the volume of the box.
1
