# Tagged Questions

Convex Optimization is a special case of mathematical optimization. It includes Linear Programming and least-squares.

222 views

### Optimization of relative entropy

Wondering if my following question is an application of information theory: Lets say we have a factory and ship boxes of stuff outside. If a competitor stands outside my factory, observes the stream ...
287 views

### Subgradient of convex minimization duality

$$\min(f_0(x))$$ $$\text{s.t. }f_i(x) \le y_i \forall i, i = 1 ,\ldots, m$$ $$f_i : \text{convex};\quad x : \text{variable}$$ It is also considered that $g(y)$ is the optimal value of the problem ...
191 views

### Need advice: what should be my next step?

I am dealing with a quite algebraic question and I arrived at some good point. I had $2$ equations with $2$ unknowns and I was able to eleminate one of the variables. My final equation still seems ...
127 views

### Conditions under which the damped Newton method is globally convergent?

Consider the problem of minimizing a convex function over $\mathbb{R}^n$ \begin{align} \min_{x\in\mathbb{R}^n}f(x) \end{align} Consider the damped Newton method (from Nesterov's book Introductory ...
199 views

### How to find accuracy of Matlab's quadprog solver?

I have solved with quadprog from Matlab a strong convex quadratic problem given as $$f(x) = x^TQx + c^Tx$$ with constrains $$Cx \leq b.$$ Now the output of quadprog is: Minimum found that ...
120 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 coordinate-wise Lipschitz constant $M$ (meaning that for all ...
93 views

### the dual of the dual is the primal?

Consider a convex optimization problem (call it $P$). Consider its dual (call it $D$). Is it true that the dual of $D$ is $P$? For linear programming, it is true. I'd just like to know under which ...
111 views

### Primal-Dual pair in SDP

Let's we have a primal model like $\max~~ x + \langle I, Z \rangle$ $s.t. ~~~Ax + y I - Z \preceq B$ $~~~~~~~~~Z \succeq 0, ~X \geq 0, ~~y ~free$ where $A, B \in {\mathbb R^{n \times n}}$. The ...
122 views

### When can the optimal value of a SDP be achieved?

Looking at semidefinite programs, are there any sufficient conditions for the solvability (i.e. the optimal value can be achieved, that is infimum=minimum)? Obviously if the problem is unbounded, the ...
35 views

### Reference for elementary result in optimization

Let $U(\mathbf{z})$ be a convex, twice differentiable function, and $F(\mathbf{z},\mathbf{q})$ be convex and twice differentiable separately in $\mathbf{z}$ and $\mathbf{q}$. Consider the problem of ...
93 views

### Variational characterization of nuclear norm

The nuclear norm $||\cdot||_{*}$ of a matrix is defined as the sum of its singular values. Working from the result at the bottom of this blog post, we have, for a matrix $\mathbf{X}$ and its ...
100 views

27 views

### Maximum overlap of two convex bodies

Let $\Delta_n=\big\{(x_1,x_2,\ldots,x_n): \sum_i x_i \leq 1, x_i \geq 0\big\}$ denote an n-dimensional simplex. I am trying to find an $y \in \mathbb{R}^n$ such that the $y-$shifted negative simplex ...
48 views

### Center of mass of vertices without enumeration?

Given a $n$-dimensional convex polytope defined by $A x\leq b$ and $A_{eq} x = b_{eq}$, is there an efficient way to determine the average coordinates of all vertices without enumerating them? (As if ...
45 views

249 views

109 views

### The importance of the full-row-rank assumption for the simplex method

Consider a linear programming model in the usual form ready for applying the simplex method. I understand that having the constraint equations' coefficient matrix $A$ be of full row rank means not ...
160 views

### Does convexity of a function guarantee tractability of finding its minimum?

Formulating a problem as a convex optimization problem usually implicitly considered to imply being able to find global minimizer of the objective. My question is that if it is true or not. ...
556 views

### SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through Lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...
123 views

### On the duality gap for quasiconvex optimisation problems

This stack exchange question got me thinking about quasiconvex analysis. Given a compact,convex subset $X\subset \mathbb{R}^n$ and a quasiconvex function $f:X\rightarrow \mathbb{R}$ Define the ...
348 views

### How to show that a function is piecewise linear

Let z(t) = min $(c+t d)^T x$ s.t $Ax <= b$ Show that Z(t) is a concave, piecewise linear function of t. I'm really not sure how to even start proving this, I would really ...
477 views

### sufficient condition for KKT problems

For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that: "The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
366 views

1k views

31 views

### Sup of a linear function

Let $X$ be a banach space or simply a normed space and $C$ a convex (closed) subset of $X$. It is true that if $x \in C$ is such that $f(x)=\sup f(C)$, (in other words $x$ is a supporting point for $C$...
41 views

### Optimizing potential

Let $X$ be a vector field on a Riemmanian manfiold $M$. I recently read that solving: $$\operatorname{argmin}_{\phi}\int_M (\nabla \phi - X)^2 d\mu,$$ where $\mu$ is ...
92 views

29 views

### Theoretically understanding the algorithmic solution to this Quadratically Constrained Quadratic Optimization Problem

I have a simple QCQP problem to solve: $\min_{t} x(t)^{T}Ax(t)$ subject to constraints $x(t)^{T}Ax(t) > 1$ where A is a positive definite matrix and $x(t) \in \mathbb{R}^2$ is some time ...
20 views

### Duality between $Ax<b$ and another system, but Gordan's just misses

I am trying to show that $$Ax < b$$ Is feasible iff $$A^T y =0 , b^Ty + s = 0, (y,s) \ge 0, (y,s) \ne 0$$ Is infeasible. Work So Far Now when I try to hit this with Gordan's Lemma I seem ...
32 views

### Is it possible to use regularization to minimize the (expected) number of non-zero digits in a number?

This question may be slightly related to this question on length of the representation of a number in a certain basis. Introduction / Background In image and video coding, particularly the ...
24 views

### Can a subgradient always be found in polynomial time?

Given a convex function, under what conditions can we find a subgradient in polynomial time? There are easy examples such as $f$ being an supremum of a finite number of differentiable functions, but ...
29 views

### Example 2.13 Conditional Probabilities, Convex Optimization by Stephen Boyd

I am reading Stephen Boyd's Convex Optimization. I can't understand the example below: what is the "convex set of joint probabilities" in this example? what is the convex set $C$ here?
I want to deal with a convex constraint \begin{align} F(P)=P^{H}AP_{0}+P_{0}^{H}AP-P_{0}^{H}AP_{0}\succeq 0 \end{align} where $(\cdot)^{H}$ represents Hermitian transpose, $A$ is a positive definite ...
This SDP problem with the variable $X \in \mbox{S}^n$ where $\rho \gt 0$ $$\max \mbox{Tr}(AX) - \rho \mbox{1}\lvert X \rvert \mbox{1} \\ \mathrm{subject\; to}\;\mbox{Tr}(X)=1, \\ X \succeq 0$$ ...