# Questions tagged [convex-optimization]

Convex minimization, a subfield of optimization, studies the problem of minimizing convex functions over convex sets. The convexity property can make optimization in some sense "easier" than the general case - for example, any local minimum must be a global minimum.

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### If-else statement in cvx

I have a problem with if-else statement in cvx which causes error. I raised this issue in cvx forum and was highly recommended to ask for assistance in this community. Could you please help to fix it ...
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### How to prove this convex-optimization problem?

I am struggling with the following optimization problems. Problem 1 \begin{align}\max_{\alpha, s_1, s_2}&\quad s_1 + s_2 - \gamma (s_1 (K_1 +c_1 + s_1) + s_2 (K_2+ c_2 + s_2) + 2\alpha K) +C\\\...
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### $\nabla_y\nabla_vf(x^*)\geq0$ for any concave $f$ if and only if $y=-v$

$f:\mathbb R^3\to\mathbb R$ is an arbitrary concave function. $H$ is a plane. $v$ is a given vector on $H$. $x^*=\max_{x\in H} f(x)$ Prove that $\nabla_y\nabla_vf(x^*)\geq 0$ if and only if $y=-v$. I ...
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### How can this convex optimization problem be proved?

Consider the following maximization problems: $\max_{x} x -\gamma p(x)$ subject to $x \in \Omega_1$ $\max_{x} x-\gamma (p(x) + q(x) )+K$ subject to $x \in \Omega_2$ where $\Omega_1$ and $\Omega_2$...
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### Quasi-convex function must be “partially monotonic”?

$f(x)$ is quasi-convex, $$x^*\in\arg\min_{x\in C}f(x).$$ How to prove that, for any $a\in C$, $f(x)$ is weakly monotonic in the direction of $(x^*-a)$? Is this simple result a part of an ancient ...
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### Linear functions in Lenstra's algorithm

I had asked this question at MathOverflow and was pointed here. I'm working on implementing Lenstra's algorithm. At the bottom of p.5 (at "construct $n+1$ linear functions"), he says to ...
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### Convexity of the variance of a mixture distribution

$X$ is a random variable that is sampled from the mixture of uniform distributions. In other words: $$X \sim \sum_{i=1}^N w_i \cdot \mathbb{U}(x_i, x_{i+1}),$$ where $\mathbb{U}(x_i, x_{i+1})$ denotes ...
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### Difference between exploration and exploitation in Simulated Annealing algorithm

In evolutionary algorithms, two main abilities maintained which are Exploration and Exploitation. In Exploration the algorithm searching for new solutions in new regions, while Exploitation means ...
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### Can we use reinforcement learning and convex optimization to solve an optimization problem?

For an optimization problem, there are multiple-type variables should be optimized. Can we use the convex optimization method to solve a subproblem of partial variables, and then, with the obtained ...
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### Relationship between extreme points and optimal solutions of SDPs

Consider this to be our SDP problem: Minimize $\langle C, X \rangle$ such that $\langle A_i, X \rangle \ge b_i$ for all $i \in [m]$ and $X \succcurlyeq 0$. For SDPs, is there a relationship between ...
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### Conditions required for strong duality to hold for SDPs

According to Wikipedia, strong duality holds when "the primal optimal objective and the dual optimal objective are equal." What are the necessary conditions for strong duality to hold in ...
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### Can every convex problem use Lagrangian dual method?

If not all constraints satisfy equalities, does Lagrangian dual method make sense to a convex problem?
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### Is a convex or MILP (without big-M) formulation possible for this problem

Assume we are given a directed acyclic graph (DAG) $G(V, A)$, where $|V| = n, |A| = m$, and the graph contains a source node $\mathbf{s}$ (i.e. every node in $V \backslash \mathbf{s}$ is connected by ...
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### DCP representation of a convex quotient of affine functions

I am trying to represent the following inequality: $$\frac{x}{1-x} \leq y \qquad\mathrm{with}\qquad 0<x<1$$ The function on the left is convex (its second derivative is always positive over ...
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### Find a point inside non-empty difference of ellipsoids

Given two ellipsoids \begin{align}\mathcal{E}_1 &= \{ X \mid X^\top A_1 X + 2B_1^\top X + C_1 \leq 0\}\\\mathcal{E}_2 &= \{ X \mid X^\top A_2 X + 2 B_2^\top X + C_2 \leq 0\}\end{align} are ...
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### How to check for convexity of the inequality constraint $−x^2+y−1\ge0$ for a minimization objective function?

I checked the Hessian which is $\begin{bmatrix}-2&0\\0&0\end{bmatrix}$ which is negative semidefinite but according to the sketch of the function it is convex. What am I missing?
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### Approximation methods for a mixed integer convex optimization problem

I have a convex objective function, e.g., minimizing the negative entropy function. My constraints are also linear. The only issue is that I also have binary variables. I am currently aware of AIMMS'...
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### Which solver solves PSD constrained convex non-linear problem

I have a problem with a vector variable $w \in \mathbb{R}^n$ and a symmetric matrix variable $V \in \mathbb{R^{n \times n}}$. I am solving a problem which is roughly like: \begin{align} \begin{array}{...
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