convex optimization – Convexity of the Hamiltonian: optimal control

Consider $$H(x, p)=sup _{u in U}{-f(x, u) cdot p-f^0(x, u)}$$
This is a classical Hamiltonian coming from an infinite horizon control problem of ODE’s with state equation, i.e.
y^{prime}(t)=f(y(t), u(t)), t>0 \
y(0)=x in mathbb{R}^n

where $u in mathcal{U}={u:(0,+infty) rightarrow U$ measurable$}$, $U subset mathbb{R}^n$

and the problem is to minimize
J(x, u)=int_{0}^{+infty} e^{-lambda t} f^0(y_x(t), u(t)) d t

Assume for simplicity that $f,f^0 in C^1_B$ and $f$ is lipchitz uniformly in $u$

Now in (Bardi, Martino, and Italo Capuzzo-Dolcetta. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Springer Science & Business Media, 1997.) it is claimed that for fixed $x$ the function $H(x,p)$ is convex in $p$.

How do you see that?

Convexity of a log quadratic function

My objective function is $log_2(x^2+x+1)$. Is this a quasiconvex function? If not, is it possible to rewrite it as a convex function?

functional analysis – Logarithumic convexity of norm of $W^{s,p}(Omega)$ space in $s$ analogue to that of $H^s(Omega)$.

I have seen following logarithmic convexity property:

Let $s$ and $s’$ be two real numbers such that $uin H^{{max{s,s’}}}(mathbb R^n)$ then for any $0le tle 1$
$$|u|_{ts+(1-t)s’}le C|u|_s^{t}|u|_{s’}^{1-t}$$

I am wondering is similar result hold in case of Sobolev space $W^{s,p}(mathbb R^n)$.

Any help or reference will be greatly appreciated.

convex analysis – Convexity and linear optimization

Consider the optimization problem $min_{xin S}f(x)$ where $f(x)=max_{i=1,…,m}{a_ix+b_i}$ and $S$ is a polyhedron contained in $R^n$.

First I want to show that the function $f$ is a convex function.

What I have done so far: We know $max$ function is convex and $a_ix+b_i$ is linear so it is also convex, hence $f(x)$ is convex. Now I am stuck at the $min_{xin S}f(x)$ part. I am not sure how I can handle the $min$ function with $S$ region.

Second I want to show that the optimization problem $min_{xin S}f(x)$ can be solved by a linear optimization problem. I don’t have any idea on how to approach this one.

Any helps would be appreciated!

economics – Microeconomics Doubt about Convexity and Satiation [$u(x_1,x_2)=-x_1^2-2x_2^2+2x_1x_2-10x_1+40x_2$]

I am having some difficulty with this question: $$text{Prove that } u(x_1,x_2)=-x_1^2-2x_2^2+2x_1x_2-10x_1+40x_2 text{represents a strictly convex preference and has a global satiation point}$$

I really don’t know how to approach this problem. For the convex part, I think I need to prove that $u(x_1,x_2)$ is strictly quasi-concave, but I don’t know how to do that.

Thank you very much.

Is Lagrangian relaxation convexity optimization problem or not?

We know that for a regular maximization LP problem, it should be
$$z^* = max_x c^Tx s.t. x in X, Ax leq b$$
where $b in mathbb{R}^m$. There is a technique called Lagrangian relaxation, which can make the problem easier to solve. The Lagrangian relaxation is given by
$$z(u) = max_x c^Tx + u^T(b-Ax) s.t. x in X$$
The Lagrangian relaxation $z(u)$ can provide an upper bound on $z^*$ for any $u geq 0, u in mathbb{R}^m$. Therefore, the tightest possible Lagrangian relaxation is
$$min_{u geq 0}z(u)$$
It is a good upper bound of the original solution. But the textbook also mentions that the above minimization problem (in u) is a convex optimization problem. Could someone explain why $z(u)$ is a convex function of $u$? Thank you very much.

On the mean value theorem and convexity

Let $f$ be a function of class $C^2$ on $(a,b)$ such that $f” ge k$ for $kinmathbb{R}$.

Show that for all $x in (a,b)$, we have $frac{f(x)-f(a)}{x-a} le frac{f(b)-f(a)}{b-a}-frac k 2 (b-x)$.

Conversely, if this inequality holds for all $(c,d) subset (a,b)$, do we have $f” ge k$ ?

matrices – Convexity of set bounded positive semidefinite matrix

can anybody help me with this question?
Let $Ain R^{nxn}$ a positive semidefinite matrix and $alpha geq 0 $ I want to prove the following:

$ M_{alpha} = { x in R^n | x^{T}Ax leq alpha}$ is convex?
to prove that I am adviced to use the following tip after proving it
for any $ lambda,mu in R $ the following holds
$lambda^2x^{T}Ax+2lambdamu x^T A y+ mu^2 y^{T}Ay$

Please excuse me for the bad writing

Deciding convexity of univariate function

In general, it’s NP-Hard to decide a convexity of a function. Is it also NP-hard to decide whether a generic univariate (one dimensional) function is convex or not?

convex analysis – Property of a function on a cone implies convexity?

Let $E$ be a Hilbert space with inner product $langlecdot,cdotrangle:Etimes Etomathbb R$ and let $C$ be a subset of $E$ such that for any $xin C$ and $yin E$, $y$ is in $C$ if and only if $langle x,yranglegeq 0$. Now $C$ is a cone since if $xin C$ then clearly $a x$ is also in $C$ for $ageq 0$.

Let $f:Cto (0,infty($ be a function satisfying the following property

If for some $x,yin C$ we have that, for all $zin C$, $langle x,zrangleleq langle y,zrangle$ then $f(x)geq f(y)$.

I have a feeling that this forces $f$ to have some kind of convexity, so here is my question :

Does that guaranty that the level sets $L_{leq f(y)}={ xin C:f(x)leq f(y) }$ for all $yin C$ are convex sets ? Are they closed ?

And what about the level sets $L_{geq f(y)}={ xin C:f(x)geq f(y) }$ ?

Let $x,y,zin C$ such that $f(x),f(y)leq f(z)$ (respectively $geq$) and with $lambdain(0,1)$, let $u=lambda x+(1-lambda)y$. Suppose toward contradiction that $f(u)>f(z)$ (resp. $<$). This implies by the aforementioned property that there is $vin C$ such that $langle z, v rangle>langle u,vrangle=lambda langle x,vrangle+(1-lambda)langle y,vrangle$ (resp. <). It feels like the only useful information that this gives is that either $langle z, v rangle>langle x,vrangle$ or $langle z, v rangle>langle y,vrangle$ (resp. $<$) and without of loss of generality we can assume it is true for $x$. I am not sure if this can lead to anything useful, but I would be grateful if some could provide me with a counter example or a proof.