Articles of convex analysis

Convex Optimization: do Primal Dual methods need to start with strictly feasible point?

I’m learning about Primal-Dual interior point algorithms from Boyd & Vandenberghe Convex Optimization, ch.11.7. Now the text mentions: In a primal-dual interior-point method, the primal and dual iterates are not necessarily feasible. However, when describing the algorithm more, it is stated: Now consider the Newton step for solving the nonlinear equations rt(x, λ, ν) = […]

Two fundamental questions about convexity of a function (number1)

The first question is as follows (see the second one): If $f$ is a convex function it is know that there is at most a single minimum. However the argument of the minimum is not guaranteed to be unique. For example $f(x)=c$ is a constant function, therefore it is convex and also concave. Its minimum […]

a convex function on a 2 dimensional closed convex set

Let us say I have a closed compact convex set $\mathbb{S}$ on the 2-D plane (eg: a circle). Let any point $p$ in the 2-D plane be represented by $p=(x,y)$. I define the max function over 2-D plane \begin{align} f(p)&=\max{(x,y)} \\ &= \begin{cases}x & \text{if $x\geq y$} \\ y & \text{if $x<y$}\end{cases} \end{align} Eg: At […]

uniqueness of Hahn-Banach extension for convex dual spaces

Let $X’$ be strict convex, i.e. for all $x_1′,x_2’\in X’$ with $\|x_1’\|_{X’}=\|x_2’\|_{X’}=1$ the implication $$\left\|\frac{x_1’+x_2′}{2}\right\|=1\Rightarrow x_1’=x_2’$$ holds. In this case the Hahn-Banach-extension is unique. I am trying to figure out how I can show this. The Hahn-Banach theorem says that for a subspace $U\subset X$ of a normed space $X$, there exists an extension $x’\in […]

Product of linear and convex function

More specific, how many maxima are there for product of these two functions: $ f(x) = ax + b $, and $ a > 0 $ $ g(x) $ is (strongly) decreasing convex function, $ \lim_{x\rightarrow\infty} g(x) = 0 $, and it is positive on $ [-\frac{b}{a}, \infty) $ on interval $ [-\frac{b}{a}, \infty) $. […]

Segment ordered density conjecture revisited

I have a set $S\subset\mathbb {R}^2$ with the following property (P) $\forall x,y\in S$, $\forall\mathscr{C}$ a convex set that contains $x$ in its interior, $bd\mathscr{C}\cap [x,y]\subset \overline{bd\mathscr{C}\cap S}$. Here $[x,y]$ denotes the segment with end-ponts $x$, $y$, $\ bd\mathscr{C}$ denotes the boundary of $\mathscr{C}$, and “$\overline{\ \ \ \ \ }$” stands for closure. In […]

Hessian matrix for convexity of multidimensional function

To prove that a one dimensional differentiable function $f(x)$ is convex, it is quite obvious to see why we would check whether or not its second derivative is $>0$ or $<0.$ What is the intuition behind the claim that, if the Hessian $H$ of a multidimensional differentiable function $f(x_1,…,x_n)$ is positive semi-definite, it must be […]

Epigraph of closed convex hull of a function

$\newcommand{\co}{\overline{\operatorname{co}}}\newcommand{\epi}{\operatorname{epi}}$ Let $X$ be an n.v.s and $f: X \to \mathbb{R} \cup \{+\infty\}$ and define $$\co f(x) \doteq \sup_{\substack {x^* \in X^* \\ r \in \mathbb{R} \\ \left \langle x^*, \cdot \right \rangle – r \leq f}} \left \{ \left \langle x^*,x \right \rangle – r \right \}$$ In other words, $\co f $ is […]

Number of local minimums and nonconvexity

I came across the following in my reading, and I like to know why this is true. “$\dots$ but, the fuction $F:\mathbb{R}^n \to \mathbb {R}$ is nonconvex since it has several local minima $\dots$” What does the number of local minima have to do with convexity?

Boundedness condition of Minkowski's Theorem

Statement: “Let L be a lattice in $R^n$ and $S\subset R^n$ be a convex, bounded set symmetric about the origin. If $Volume(S) > 2^ndet(L)$, then S contains a nonzero lattice vector. Moreover, if S is closed, then it suffices to take $Volume(S) \geq 2^ndet(L)$.” I am wondering if the condition for boundedness can be relaxed. […]