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APM462 Assignment

创建日期：2024-07-15 15:57:26

所属分类：数学mathematics

标签： APM462

Assignment

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APM462: Midterm Project
Submit your solutions to the homework on Crowdmark.
(1) Let C ⊂ Rn be a nonempty closed convex set and a ∈ Rn a point outside of
C. Suppose x0 ∈ C is the projection of a onto C, that is |x0 − a| ≤ |x− a|
for all x ∈ C. Prove that x0 is unique.
(2) Let D = {(x, y) ∈ R2|x2 + y2 ≤ r} ⊂ R2 and let ∂D denote its boundary.
Let d : D → R be the distance from a point a ∈ D to its boundary:
d(a) = min
x∈∂D
|a− x|.
Prove that d is a concave function on D.
(3) Let C ⊂ Rn be the non negative orthant (C = {(x1, . . . , xn) ∈ Rn|xi ≥
0, i = 1, . . . , n}) and f : Rn → R be convex. Consider the following
problem
minimise f(x)
subject to x ∈ C
and assume that it has a minimiser. Prove that if there exists some y ∈ C
such that there is some v ∈ ∂f(y) with v ∈ intC, then the set of minimisers
to the problem is bounded.
(4) Consider the set C = {(x, y) ∈ R2|x2 + y2 ≤ 1, y ≥ 0, y ≤ x}. Let δC(x, y)
be its indicator function. Let (x0, y0) be a boundary point of C. Find
∂δC(x0, y0).
(5) Consider the following problem:
minimise f(x, y) = (x− 1)2 + (y − 3)2
subject to g(x, y) = |x|+ 2y − 2 ≤ 0
(a) Solve the problem using subdifferentials.
(b) Convert this problem into an equivalent minimisation problem which
you can apply the KKT conditions to. Then use the KKT FONC to
find candidates for minimisers. Do you get the same candidates as in
part a?
(6) Consider the following problem:
minimise f(x, y) = 3x2 − 2y
subject to g(x, y) = |x+ 2|+ |y − 3| − 1 ≤ 0
(a) Find ∂f(x, y) and ∂g(x, y).
1
(b) Solve the problem using subdifferentials.
(7) Consider the following problem:
minimise f(x, y) = max{|x|, y + 2}
subject to g1(x, y) = 2x
2 + y2 − 1 ≤ 0
g2(x, y) = |x| − 2y − 1 ≤ 0
(a) Find ∂f(x, y), ∂g1(x, y), and ∂g2(x, y).
(b) Solve the problem using subdifferentials.
The following 3 problems—the ones indexed by Roman numerals—are
for practice only and are not to be turned in.
I. Let x = (x1, . . . , xn) ∈ Rn. The ℓ∞ norm of a vector is defined as follows:
|x|∞ = max
1≤i≤n
|xi|.
Let f(x) = |x|∞. Find ∂f(0).
II. Let f : Rn → R be convex and C1. Prove that ∂f(x) = {∇f(x)}.
III. On the interval [1, 3], approximate the function x2 by the affine function
ax+ b such that the quantity
sup
x∈[1,3]
|x2 − (ax+ b)|
is minimised.

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239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 MGMTMFE 405 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