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Mathematics Stack Exchange

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
7419 questions
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Convex function wrt. an argument

I have problem formulation saying $ q:X\times W \rightarrow \mathbb{R} $ is convex wrt. its second argument $\forall\,x\in X$. How is different from saying q is convex? And does this mean $\frac{\partial^2q(x,w)}{\partial w^2}\geq 0 \,…
1233023
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Strictly seperating convex sets if one of the two is bounded?

It is mentioned that we can only separate two convex sets only when at least one is bounded (compact). Now if we take epigraph of $f(x) = x^2 +1$ and the other is below the $x$ axis, I can find a function $f(x)=0.5$ that can separate these two…
Arun Chauhan
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Strict convex function?

I try to prove that $g(x)= K |x|^2/2 + z(x)$ is strictly convex, given that $z(x) \geq - m(1 + |x|^p)$ with $m \geq 0$, $0 \leq p \leq 2$, forall $x \in \mathbb{R}^n$, provided $K$ is sufficiently large. How to show that?
AlexisZorbas
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Distance of point, x from closed convex set C when x is not in C.

In the given Lemma. Suppose $x$ is a point outside a non-empty closed convex set $C$ . Then there exist a unique point $C(x)$, on the boundary of $C$ , which is closer to $x$ than any other point in $C$ . What I am not understanding that why the…
Arun Chauhan
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1 answer

Optimization using KKT

Optimization of: $x_{1}^{2}+x_{2}^{2} + x_{3}^{2}$, given that $x_{1}-x_{2}\leq0$, $x_{2}-x_{3}\leq0$, $(x_{1} - x_{2})^{2} + (x_{2} - x_{3})^{2} = \delta$ and $x_{1} + x_{2} + x_{3} = 0$. I tried using KKT condtion but not getting the solution.
Satya Prakash
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Distributed optimization for convex problem with coupling affine constraints.

I have the following convex optimization problem $$ \begin{split} \min_{\tilde{q}}\; & ||\tilde{q} - r||_2^2\\ s.t. & A^T\tilde{q} \leq b \end{split} $$ where $\tilde{q} = [q_1;...;q_N]$ is vector that has subvectors $q_i$ inside. and I would…
Mykola Servetnyk
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Regarding inequality constraints in convex optimization

Do variables in a convex optimization problem have to satisfy the inequality constraints only at the optimum or throughout the iteration from initial point to convergence? Mathematically, it doesn't matter but if those variables were variables of a…
Sebin Mathew
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maximum of a multivariate function on the boundary

I have a smooth function of 2 variables: $F(x,y)$ with domain $D$, which is a compact set. If for every fixed $y_0$, $F(x,y_0)$ is convex, then can I claim the maximum of $F(x,y)$ is on the boundary of $D$? not interior point of $D$?
kim kevin
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Optimization of a locally convex function

I have devised some function that I wish to minimize. Overall the function is not convex, but it is so on the relevant constraint set of its arguments (which is also convex). My question is now whether the overall nonconvexity of the function may…
Andreas Steimer
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Conjugate gradient method on quadratic function

Consider the quadratic function $\mathbf{\frac{1}{2}x^TGx+b^Tx}$ in four variables where $$\mathbf{G}= \begin{bmatrix} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end{bmatrix}$$ and $\mathbf{b}=(-1,0,2,\sqrt{5})^T$. Apply…
ozarka
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2 answers

Strict convexity of a function

How to prove that for each $\alpha > 1$ the function $\psi(t)=|t|^\alpha$ is strictly convex on the real line?
roc11111111
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Minimize the sum of two convex functions s.t. linear constraint, when we know the solution to each

Suppose $x$ is $n$-dimensional real, and $f(x)$ and $g(x)$ are real and convex R1. I want to find $x$ to minimize $Q= f(x)+g(x)$ s.t. $a’x=b$, for $a$ $N \times 1$ real vector and constant b. Let's call this solution $X^*$. How is $X^*$ related to…
MH-NY
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When does $\ell_1$-norm regularization yield the same result as $\ell_0$-norm regularization?

Often we would like to solve an optimization problem such as $$ \text{minimize} \quad f(x) + \alpha \| x \|_0, $$ where the optimization variable is $x \in \mathbb R^n$, $f:\mathbb R^n \to \mathbb R$ is convex, $\alpha > 0$, and $ \| x \|_0$ is the…
eternalGoldenBraid
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How can I prove the convexity of a function through this?

$(\nabla f(x) - \nabla f(y))^T (x-y) \geq 0$ $f$ be twice differentiable, with $\mathrm{dom}(f)$ convex. I can derive the convexity of $f$ from the above inequality through first order condition, but is that also the sufficiency for convexity of…
good2know
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How to examine the convexity of this function?

$x$ is a vector that $x=(x_1,\ldots,x_n)$ $f(x) = -\sum_{i=1}^n \log(x_i)$ I know this is equivalent of proving that $\Pi x_{i}$ is a convex set or not, but how to prove this step? Since it seems that I can't write derivative from this, or there are…
good2know
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