0
votes
1answer
104 views

Eigenvalues of a given parametrized matrix.

Let $\mathbf{A}$ and $\mathbf{B}$ be two complex rank-one $N\times N$ positive semi-definite matrices. Let the matrix $\mathbf{C}$ be defined as \begin{align} ...
1
vote
2answers
209 views

On a version of gradient descent

I am trying to read this paper and have gotten stuck. The author considers the problem of minimizing a convex function whose gradient has Lipschitz constant $M$ and considers the scheme $$ x(t+1) = ...
0
votes
1answer
159 views

(probably simple) optimization question

Suppose you have a concave function defined over a non-polyhedral convex cone and you are interested in the infimum. What would be standard approaches to tackle the question? (The cone is actually PSD ...
4
votes
1answer
273 views

Going in the direction of the gradient

First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$. Now my ...
2
votes
1answer
156 views

Lipshitz Constant of the convex extension of a submodular function

The title says it all :) Given a submodular function (take the rank function of a matroid, for a concrete example) $f:\{0,1\}^n\rightarrow \mathbb{R}$, we can extend it to a convex function ...
0
votes
0answers
224 views

A question regarding Danskin's theorem

Suppose $\phi({\bf x},{\bf z})$ is a continuous function of two arguments, $\phi: {\mathbb R}^N \times Z \rightarrow {\mathbb R}$, where $Z \subset {\mathbb R}^m$ is a (non-empty) compact convex ...
1
vote
1answer
308 views

conjugate function for matrix mixed norm

I am familiar with the conjugate function of the vector norm, which uses the concept of dual norm and is defined as follows: ...
2
votes
1answer
224 views

Generalized unique nearest point problem for a compact, convex set in a strictly convex Banach space.

If $X$ is a Real Banach space with strictly convex norm, it is known that for any non-empty compact, convex set $K$ and point $x_0\notin K$, there exists a unique point $z_0\in K$ minimizing the ...
2
votes
1answer
1k views

Sufficient conditions for gradient descent convergence

I have an unconstrained optimisation problem with convex objective function $f(x)$. Suppose I have access only to some function of the gradient $\hat{\nabla}= g(\nabla f)$, and I take gradient steps ...
3
votes
2answers
624 views

Computational complexity of unconstrained convex optimisation

What is known about the relationship between unconstrained convex optimisation and computational complexity? For example, for which optimisation problems and which gradient descent algorithms is one ...
2
votes
1answer
393 views

minimizing functions over simple matrix inequalities

I'm wondering if anything is known about minimizing convex, not necessarily linear functions subject to "simple" matrix equalities. To be precise, consider the following example: $min \Sigma x_i ln ...
4
votes
0answers
160 views

Is it possible to solve the argument maximization problem $\arg\max_x \langle x,l \rangle −f_1(x)−f_2(x)$ via convex duality?

I am attempting to solve the argument maximization problem $$\arg\sup_x \{ \langle x,l \rangle − f_1(x)−f_2(x) \} \ \ \ \ \ \ \ \ \ \ (1)$$ where the functions $f_1$ and $f_2$ are concave but ...
2
votes
1answer
243 views

Linear and Isometric Automorphism Groups of the PSD Cone

Let $S_+$ be the cone of psd matrices ($n\times n$ real symmetric positive semidefinite matrices). This cone is a metric space induced from the inner product $\langle A,B\rangle = tr (AB)=tr(BA)$. ...
2
votes
1answer
704 views

How do I optimize over (or take derivative wrt) a square diagonal matrix?

Hello. I'd like to solve the following optimization problem. $P_i$ is a 6x6 matrix $X$, $Y$ is a 6xk matrix $w_i$ is a kx1 vector $diag(w_i)$ is a square diagonal matrix with diagonal entries equal ...
3
votes
2answers
409 views

Is a solution of a linear system of semidefinite matrices a convex combination of rank 1 solutions?

The cone of symmetric positive semidefinite $n\times n$ matrices is the convex hull of rank $1$ matrices. That is, every symmetric positive semidefinite matrix is a convex combination of rank 1 ...
4
votes
3answers
2k views

Minimize trace of inverse of convex combination of matrices.

Hello! (First question--please forgive me if its unclear.) I am interested in efficient/approximate optimization techniques for minimizing a norm of a convex combination of symmetric, positive ...
3
votes
2answers
476 views

A lower bound of a particular convex function

Hello, I suspect this reduces to a homework problem, but I've been a bit hung up on it for the last few hours. I'm trying to minimize the (convex) function $f(x) = 1/x + ax + bx^2$ , where ...
3
votes
1answer
647 views

Lipschitz properties of minima/minimizers of convex functions of two variables

Suppose I have a function $f(x,y)$ from $\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ that is convex in both $x$ and $y$. Set $g(y) = \min_{x} f(x,y)$ What I would like is for $g(y)$ to be ...
3
votes
3answers
570 views

Some questions about Invexity

Recently, I am looking into a non-convex optimization problem whose points satisfying KKT conditions can be obtained. Then the problem becomes how to decide whether the KKT conditions are sufficient ...
2
votes
1answer
269 views

Can subgradient infer convexity?

It is known that If $f:U→ R$ is a real-valued convex function defined on a convex open set in the Euclidean space $R^n$, a vector v in that space is called a subgradient at a point $x_0$ in $U$ if for ...
0
votes
3answers
522 views

a different algebra/representation for convex sets

Hi, I was dealing with finding a feasible region for a set of norm inequalities and the feasible region is convex. The question is not about how to find the feasible region but how to represent the ...
19
votes
5answers
10k views

Is all non-convex optimization heuristic?

Convex Optimization is a mathematically rigorous and well-studied field. In linear programming a whole host of tractable methods give your global optimums in lightning fast times. Quadratic ...
0
votes
2answers
2k views

If a quadratic form is positive definite on a convex set, is it convex on that set?

Consider a real symmetric matrix $A\in\mathbb{R}^{n \times n}$. The associated quadratic form $x^T A x$ is a convex function on all of $\mathbb{R}^n$ iff $A$ is positive semidefinite, i.e., if $x^T A ...