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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1answer
45 views

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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0answers
27 views

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 ...
5
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1answer
110 views

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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2answers
611 views

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 ...
3
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2answers
97 views

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 ...
2
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1answer
53 views

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 ...
4
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1answer
65 views

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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0answers
62 views

Can every convex problem use Lagrangian dual method?

If not all constraints satisfy equalities, does Lagrangian dual method make sense to a convex problem?
2
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1answer
91 views

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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0answers
75 views

Question on quadratically constrained quadratic program

If the constrained optimization problem is a quadratically constrained quadratic program of the form \begin{align}\min&\quad x^HQx-a(x+x^H)+b|z^Hx|^2\\\text{s.t.}&\quad\|x\|^2\le1\end{align} ...
3
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2answers
112 views

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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2answers
136 views

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 ...
4
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1answer
57 views

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?
3
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1answer
136 views

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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1answer
104 views

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}{...
4
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1answer
80 views

Minimize a convex function over a sphere

Problem description Let $\mathcal{C} = \{X \in \mathbb{R}^n \mid g(X) \leq 0\}$ with $g(X)$ a convex function. Suppose I need to solve the feasibility problem, for a given $r>0$ $$ \exists ^?X \...
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0answers
58 views

Strong Duality and Slater Condition

I am studying the Duality Chapter of Convex Optimization by Boyd. Is it possible that strong duality holds for non-convex optimization? If yes, is there any specific condition? And, what is the ...
6
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2answers
134 views

Existence of Optimal Solution

Assume we are solving $\min\{f(x) \ | \ x \in S \}$. If $f: \mathbb{R}^n \mapsto \mathbb{R}$ is a proper closed convex function, and $S$ is a non-empty closed convex set, does this imply that the ...
4
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1answer
148 views

Cutting-planes application procedure for a specific problem

Sort of following up with this question. I reformulated another model to make it convex and possibly solve it with some cut generation method. I would like to double-check whether I am doing it ...
7
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1answer
79 views

Ridge Regression lagrange duality

In every machine learning book we see that it is roughly mentioned that the ridge regression: $$p_1^* = \min\limits_{\beta} \ \left( \mathrm{RSS} + \lambda\sum_{j=1}^p \beta_j^2 \right)$$ is ...
6
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0answers
50 views

Semi-definite Programming, non standard notation

The usual way to define a semi-definite program (SDP), e.g., as given in Boyd and Vandenberghe's convex optimization book, is: $$ \begin{array}{cl} \min & c^\top x \\ \mathrm{s.t.} & 0 \succeq ...
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0answers
182 views

Convexity of the projection of a convex set

Question: A set $S \subset \mathbb{R}^m \times \mathbb{R}^n$ is convex. Using the fact that affine maps preserves convexity prove that $S(y) = \{x \in \mathbb{R}^m\mid (x,y)\in S \}$ and $\hat{S} = ...
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2answers
133 views

Convex Optimization: Separation of Cones

I am trying to solve Exercise 2.39 at Boyd and Vandenberghe's Convex Optimization book. In one source, the answer is given as: 2.39 Separation of cones. Let $K$ and $\tilde K$ be two convex cones ...
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4answers
255 views

Disciplined convex programming representation of $x\sqrt{1-x}$

Anyone have an idea for a disciplined convex programming (DCP) representation of the concave function $x\sqrt{1-x}$, which is defined over the domain $[0,1]$? The Taylor series about $x=0$ is $$x - \...
8
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1answer
232 views

Convexity/Concavity of Average Number of Jobs in M/M/1 Queue?

I am working on a problem involving the average number of jobs $L$ in an M/M/1 queue with arrival rate $\lambda$, service rate $\mu$. For traffic intensity $\rho = \frac{\lambda}{\mu}$, $$ L = \frac{\...
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2answers
278 views

Convex Maximization with Linear Constraints

I am doing active research in convex maximization w.r.t. linear constraints. There are many cases which can be efficiently approximately solved, e.g., convex quadratic maximization, log-sum-exp ...