# Tagged Questions

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

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### Be C a matrix n x n positive semi definite. proof x'Cx is convex and sqrtroot(x'Cx) is convex.

Hi I have a homework from optimization and I want to know how to do the following exercise: Be C a matrix n x n positive semi definite. proof that: (1)$x^tCx$ is convex. (2)$\sqrt{x^tCx}$ is convex. ...
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### Convert Quadratically constrained basis pursuit to LASSO

The Quadratically constrained basis pursuit is to solve \begin{align} \hat{\boldsymbol{x}} &= \arg\min \|\boldsymbol{x} \|_1 \\ s.t. & \| \boldsymbol{Ax} - \boldsymbol{y} \|_2^2 < \eta ...
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### Existence of unique maximizer in R^n

This sounds like a very basic question, but I have a hard time pinpointing the necessary and sufficient conditions... Let $f : \mathbf{R}^n \to \mathbf{R}$ be a function. I want to prove that there ...
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### Properties of convex function with Lipshitz continuous gradient (Prof. Nesterov's textbook)

I am reading the Prof. Nesterov's textbook: Introductory lectures on convex optimization - a basic course p.57 I have problem in the following: My ...
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### Proof of unique solution of strongly convex function (Prof. Nesterov Paper)

I am reading the paper of Prof. Yurii Nesterov: Primal-dual subgradient methods for convex problems I am confused about the green part of the following: $\\$ ...
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### How to prove this function is quasi-concave? [closed]

Consider the function $f(x,y) = x(1-y)\log(1+y/x^2)$, where $0\le x, y\le1$. Is this function quasi-concave?
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### Convex hull of halfspace and point is not a polyhedron

Let $S=conv(H \cup\{x\} )$ denote the convex hull of $H \cup\{x\}$ where $H \subset \Bbb{R}^n$ is a halfspace and $x\in\Bbb{R}^n, x\notin H$. I need to prove that $S$ is not a polyhedron and my ...
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### Can we solve minimax in this way?

I am working to use proximal operators for solving a minimax optimization problem. It is known that if you use alternative optimization, the algorithm cycles, see an answer to this question ...
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### How can I solve this as an optimization problem?

I would like to find x such that (Ax).^2 + (Bx).^2 == I (using Matlab syntax). A, B are matrices and I is a vector, all with real values. The number of equations is less than the number of variables, ...
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### Convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant 1$

I would like to ask the convexity of $y\geqslant\frac{1}{x}$ and $xy\geqslant1$. We know that $y\geqslant\frac{1}{x}$ is convex for $x>0$. But if we transform the inequality into ...
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### If we know the convex conjugate of $f(x,y)$, what can we say about the conjugate of $f$ in $x$?

Say $f^*(x,y)$ is the convex conjugate of $f(x,y)$. Now take $g_{y_0}(x) := f(x, y_0)$. Is there any relationship between $g^*_{y_0}(x)$ and $f^*(x, y_0)$?
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### Solving nonconvex problem by iterating convex ones

I have a convex problem with the following properties: -The energy to be minimized is convex - it is basically a norm. -The domain is defined by a set of convex cone constraints inequalities. I am ...
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### Optimality condition

I was looking at a few results of convex optimization and I'm stuck with a part of a proof. Consider the following minimization problem: \begin{align} \text{minimize} \quad &\Phi(x) \\ ...
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### Problem to prove function is convex or not?

How do I plot function $f(x1,x2)=x^4_1+x^4_2$ such that $x^2_1+x^2_2=1$ and $x_1,x_2\in(0,1)$? Does it possible to plot in MATLAB so that I can visualize the function? Also I am trying to check ...
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### Convexity, Hessian matrix, and positive semidefinite matrix

I would like to ask whether my understanding of convexity, Hessian matrix, and positive semidefinite matrix is correct. For a twice differentiable function $f$, it is convex iff its Hessian $H$ is ...
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### How to perform a quasiconvex optimization

I have a quasiconvex objective function $f:\mathbf{R}^n\rightarrow \mathbf{R}$ which I would like to minimize over a simplex $S\subseteq \mathbf{R}^n$. I have looked pretty hard but have been unable ...
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### Is a global optimal solution of a convex problem always unique?

I do not have a specific problem. Could a convex optimization problem (not strictly convex) have alternate solutions?
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### ADMM Formulation for L1 minimization with equality or inequality constraints

A simple ADMM formulation exists to minimize $||Ax - b||_1$ (L1 norm minimization): http://web.stanford.edu/~boyd/papers/admm/least_abs_deviations/lad.html How do I extend this formulation to work ...
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### Can we use simple alternating optimization for minimax (saddle point) problems?

Consider a function $f(x,y)$, convex in $x$ and concave in $y$. we are interested in the following optimization problem, \begin{align} \min_{x \in D_x} \max_{y \in D_y} f(x,y) \end{align} Because of ...
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### Boundary of a convex body

I need your help in defining a boundary for a given convex body, $P$ without knowing the shape of $P$ meaning that I can't say that $P$ is a polygon or any shape which is convex. Meaning that by ...
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### A light solution of a quadratic programming problem

I have a simple and light quadratic programming problem that I need to solve, as following: \begin{align} & \underset{x}{\arg\min} & & \dfrac{1}{2}x^T x-z^T x\\ & \text{subject ...
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### Prove a set is convex

I have problems to make proof for below two statements. Let Γ be the LP max cᵀx s.t. Ax ≤ b. prove that set of all optimal solutions to Γ is a convex set Let x' be a basic feasible solution of Γ. ...
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