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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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A question on notation in convex optimization

$\mu_E$ is a $4 \times 1$ vector composed of known constants, and $\mu$ is a vector of the same dimension but with unknown variables. Let us say $\mu = (x_1, x_2, x_3, x_4)^T$. What is the meaning of the following notation in optimization?…
cgo
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Given Lagrange Optimization

I would like to maximize $P = 2pr + 3pq + 2rq$ given that $p+q+3r=1 \wedge p,q,r \geq 0 \wedge p,q,r \leq 1.$ I was considering to develop $P$ as a function of three variables and using standard Lagrange Optimization, but I was wondering if it would…
Erica
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Prove or disprove a monotonicity property of a certain convex optimization problem

Let $R = (r_{ij})$ be an $n\times k$ real matrix with only positive entries, and consider the convex optimization problem $\max f(x) = \sum_{i=1}^n \log \sum_{j=1}^k r_{ij} x_j$ such that $\sum_{j=1}^k x_j = 1$ and $x_j\ge 0$ for all…
Jobst
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Convex Optimization

If $q>p>0$ show that the point $(x,y)=(0,0)$ minimizes the function $$f(x,y) = (y-px^2)(y-qx^2)$$ locally on the lines $$y=ht, x=kt$$ through the point (x,y).
Jackson
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Dual problem of piecewise linear function

I would like to see the geometric interpretation of the relationship between the primal problem and the dual problem on the $x,y$-plane. So I am looking at an example of minimizing the maximum of some linear functions.…
Sapphire
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The Dual Problem

I have a question related to the dual problem of a maximization primal problem with norm inequalities constraints. I want to find the dual problem for the following primal optimization problem: $\text {max} -b^Tv $ $\text{subject to }\Vert…
Ahmed Zeki
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Monotropic polyhedrally constrained programming, implementations

I'm looking for algorithm implementations that can solve problems of the type $$\min \sum_i f_i(x_i) \\ s.t. Ax \leq b \\ Cx = d \\ x_i \in [\ell_i,u_i]$$ I.e. polyhedrally constrained optimization problems with a separably convex objective function…
Benjamin Lindqvist
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Minimizing a quadratic term

$\mathbf{x_1},\mathbf{x_2}$ are known and I need to solve the following objective wrt to one variable $\mathbf{y}$. The single constraint is $y(1,1)=1$. This is expressed as an inner product $\mathbf{y^T}\mathbf{e_y}=1$ \begin{align} \mathbf{e_y} =…
NAASI
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Rewriting a strictly convex quadratic equality constraint

I have the following constraint $$ x^TAx+b^Tx=c $$ where $A$ is a positive definite matrix. Is there any way to take advantage of the strict convexity of this expression to reformulate the constraint as something simpler? Thoughts/attempt: I was…
jonem
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MLE of an increasing nonegative signal from convex optimization book

I need help solving this problem from Stephen Boyd's Convex Optimization additional exercise. Its question 6.6 from additional exercise. Maximum likelihood estimation of an increasing nonnegative signal. We wish to estimate a scalar signal $x(t)$,…
Art Zahar
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How to solve the following non-convex optimization problem?

$$\min \|X\|_{*}+u\|Ax-b\|_2^2+v\|Cx\|_2^2 + wx^THx$$ where $x$ is vec($X$), $u,v$. is constant, H is a symmetric matrix，but it is not semidefinite. Is there any software to do this? Can the software cvx solve it?
yang
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intuition behind subspace of $R^n$

Hi: I've been reading an optimization text by Charles Byrne, "A First Course In Optimization". I'm currently going through the chapter where he explains things about convex sets and convex functions and projections. The material is explained well…
mark leeds
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proof that length of difference of projections implies equality of length of normals

Hi: I'm reading a book on optimization and there is an interesting stated theorem but I don't know how to prove it. Notation: Let $P_{c}(x)$ denote the projection of onto a convex set c which is a subspace of $R^{n}$. Then, the theorem is the…
mark leeds
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MD (Mirror Descent) over l_1 simplex lower bound proof

I'm looking for a proof of a lower bound of the Mirror Descent optimization algorithm over l_1 simplex. I am not asking to reproduce a proof here, but rather for a reference. I did go over the Nemirovsky & Yudin book on the subject (A. Nemirovski…
Alex Kreimer
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Proof that $d\left(\mathbf{x},S\right)=\displaystyle\min_{\mathbf{y}\in{S}}\,\|\mathbf{x}-\mathbf{y}\|$ is convex

I'm wondering if the following is sufficient to show that for a closed, convex set $S$ $$d\left(\mathbf{x},S\right)=\displaystyle\min_{\mathbf{y}\in{S}}\|\mathbf{x}-\mathbf{y}\|$$ is convex. Definition of convexity: $$f\left(\theta\mathbf{x} +…
Phillip Cloud
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