(5) (5pt) Show that if a convex function f as two distinct global minima r, r** ina convex set K, then it has infin...
f(x) Sa} 5. Show that if f is a convex function on R" then for any value a E R the set {<ER is convex, and so too is the set {x ER" f(x) <a}.
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on C if and only if Vf(x*)-(x - x*)2 0 for all x in C.
Suppose that f(x) is a convex function with continuous first partials defined on a convex set C in R". Prove that a point x* in C is a global minimizer of f(x) on...
Convex Optimization
Let f: R R be a differentiable function on R. Show that f is convex iff f' is nondecreasing (i.e. x y f'(x) <f'(y)).
R is convex over C. Show that 4) Assume that is a convex subset of IR" and that f: the set of global minimums of over C is convex.
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
Running average of a convex function. Suppose fR R is convex, with R+ S dom f. Show that its running average F, defined as F(a)-f(t) dt. dom F-R++ 2 0 is convex. You can assume f is differentiable.
need help with all a, b, c
2. 15 Marks (a) Suppose that f : R" R is convex but not necessarily smooth. Prove that h-af is a (b) Suppose that f : R -R is convex and smooth. Also assume that f(x) > 0 for all z (c) Show that the set S = {(x,y) : y > 0} is convex and that the function f(x,y)-x2/v is convex function if a-0. Show with a simple example that this is...
help me.
5. consider set F(R):ff: f:R-R), but set all function with set real number in domain and codomain. Show "addition" in any two function it.eCE(R) to produce new function such as given: ttgR2R which is every xER such as given:(tg)lx)-fx)+g(x), and any real number k ER, multiply it with any element f EF(R) to produce new function as given: kfRR in every value xER such as given:(k:0(x):-kfx)(observe it with multiply dua real number) (a) Show. FIR) ith addition and...
Support function. The support function of a set C C R is defined as We allow Sc(y) to take on the value too.) Suppose that C and D are closed convex sets in R". Show that C D if and only if their support functions are equa
Support function. The support function of a set C C R is defined as We allow Sc(y) to take on the value too.) Suppose that C and D are closed convex sets in...
5. (a) Show that if the functions f and g are log-convex, f+g is also log-convex. Give a counter example to show that this is not true for log-concave functions (Hint: log(f +g)log(elogf +elogs). Show that this is convex by the second-order test for convexity.) (Hint: Use the definition of log-convex functions.) (Note: Harmonic mean of a,b is defined as T^T.) b) Suppose f is convex, g is non-decreasing and log-convex. Show that h(x) g(f(x)) is log-convex. (c) Show that...
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set
I. Let f : R → R be a continuous function. Show that ER sup is a Fo set