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Questions tagged [convex-optimization]

A convex optimization problem consists of either minimizing a convex objective or maximizing a concave objective over a convex feasible region.

Convex Optimization is a special case of mathematical optimization where the feasible region is convex and the objective is to either minimize a convex function or maximize a concave function. Linear Programming is a special case. Convex Optimization problems as a class are easier to solve numerically than general mathematical optimization problems.

The following problems are all convex minimization problems, or can be transformed into convex minimizations problems via a change of variables:

  • Least squares
  • Linear programming
  • Convex quadratic minimization with linear constraints
  • Quadratic minimization with convex quadratic constraints
  • Conic optimization
  • Geometric programming
  • Second order cone programming
  • Semidefinite programming
  • Entropy maximization with appropriate constraints
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Practical optimization of a convex 1D function without derivatives?

What is a "good" way to optimize a 1D convex function $f(x)$ without using the derivatives of that function ? By "good", I mean a method which exploits the function convexity and which minimizes the number of function evaluations (suppose that the…
VanillaSpinIce
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Usefulness of dual variables in ADMM

I am wondering about applications of dual variables that are used in the Alternate Direction of Method of Multipliers (ADMM). In the consensus form of ADMM they grow until they have forced consensus among variables that are treated separate for ease…
Dave31415
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The solution of a convex optimization problem is unique

There is a statement in a thesis I am reading, The solution of a convex optimization problem is unique, and the global and the local minima are essentially the same. Is there a proof for it? Why does it hold?
Gigili
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Finding the inverse image of a function

While I was reading a Convex Optimization book, I found an example, which is attached below. What is confused me is that how the authors derived/concluded that the hyperbolic cone is the image inverse of f(x).
Learner
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Why $f:R^2\to R$, with $\mbox{dom} f=R^2_+$ and $f(x_1,x_2) =x_1x_2$ is quasiconcave?

Why $f:R^2\to R$, with $\mbox{dom} f=R^2_+$ and $f(x_1,x_2) =x_1x_2$ is quasiconcave? I have tried to use Jensen eniquality to check that superlevel set $\{x\in R^2_+ | x_1x_2 \ge \alpha\}$ is convex. $$\begin{align*}(\theta x_1 + (1-\theta)…
Yola
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Why this problem is unbounded, even though it's dual is feasible?

In this paper, An Efficient Inexact ABCD Method for Least Squares Semidefinite Programming,page 2, there is a problem called, (P): \begin{align} P: &\min_{X,s} && ‎ \frac{1}{2} ‎\Vert X-G\Vert^2‎ +\frac{1}{2} \Vert s-g\Vert^2…
user85361
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Variational inequality and saddle point problem

I refer to the following paper: (fix the wrong link) https://papers.nips.cc/paper/5723-adaptive-primal-dual-splitting-methods-for-statistical-learning-and-image-processing.pdf In that paper, we consider the saddle point problem: $$ \min_x \max_y…
jakeoung
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Is the minimum point of a strictly convex function stable?

This is a problem I figured out after seeing the definition of minimum stable point and I think the following tense is true: Let $f(x)$ be a strictly convex function whose minimum values is $f^*=f(x^*)$. Show that $x^*$ is a stable minimum…
AlephZero
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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 whether $f$ is convex or not, because I am trying to…
P Pradhan
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Logarithmic Function Behaivour

I have read about Logarithmic function. We can use the second-order condition to show that the $f(x)=\log_2(1+x), x \geq 0$ is a concave function. Now, is $g(x)$ a concave function? How can I prove this fact?…
Hossein
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why Quasiconvex function is not concave?

A quasiconvex function is a function whose all sublevel set are convex. I am curious to know whether a quasiconvex function is a concave function.
user2104150
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Smoothness of total variation norm with weight

Let me write total variation norm $$ \|u\|_{TV} = \max_{z\in Q} \langle z, Du\rangle, $$ where $Q$ is the unit ball in $\mathbb R^2$ and $D$ is the corresponding gradient matrix. I can smooth TV by $$ \max_{z\in Q} \langle z, Du\rangle -…
jakeoung
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Suppose that f : X → ]−∞, +∞] is convex and proper, that x ̄ ∈ dom f and that λ > 0. Show that

Suppose that $f : X → ]−\infty, +\infty]$ is convex and proper, that $\bar{x} \in dom f$, and that $\lambda > 0$. Show that $$\partial^{\infty}(\lambda f)(\bar{x}) = \lambda\partial^{\infty}f(\bar{x}).$$ I have this property but I am still unsure…
mathjl
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Is This Constraint Convex?

The variable is $\mathbf{x}=\bigl[x_1,\ldots,x_n\bigr]^T$ and $a_k$ is given number. The constraint is the following: $$\dfrac{|a_k\cdot x_k|^2}{\sum_{j\neq k}^n|a_j\cdot x_j|^2+1}\geqslant \alpha.$$ After some modifications, we can easily get the…
drzbir
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Gradient mapping minimization problem

I am trying to solve this particular problem which can be found in this paper: $$\underset{y\in\Delta_n}{\text{argmin}}\{\langle\nabla f(x),y-x\rangle+\dfrac{1}{2}L\|y-x\|_1^2\}$$ I tried formulating the above problem as a projection…
Sapphire
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