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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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Is a the argmin of a strictly convex objective function $g( \boldsymbol{x}) $ subject to linear continuously varying constraints continuous?

I have the following problem: $$\begin{array}{ll} & \boldsymbol{x}^*(t) = \arg \min_{ \boldsymbol{x}}\text{ }g( \boldsymbol{x}) \\ \text{subject to} & \boldsymbol{A}(t)\boldsymbol{x }= \boldsymbol{B}(t)\\ & 0\leq x_i \leq x_{\max} , …
Einar U
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Operations that preserve convexity

I am trying to detemine if the expression below is concave or convex using operations that preserve convexity http://web.stanford.edu/class/ee364a/lectures/functions.pdf. $Var[x] = E(x^2) - E(x)^2$ $E(x)$ is concave / convex because it is a sum of…
Kong
  • 884
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Proving a Set Convex

I am trying to prove a set is convex. This is a problem from Boyd's Convex Optimization text. The set is $\{\hat{x} + tv | \alpha t^2 + \beta t + \gamma \leq 0\}$. This is actually an intersection of a line and a set, the line can be thought of as…
jrand
  • 1,959
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What's the relationship between quadratics and convex functions

Are all quadratic functions convex and vice versa or are quadratic functions just one type of convex function? If so, could someone please provide examples of functions that are convex but not quadratic?
miggety
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Maximum concyclic points

Given n points, find an algorithm to get a circle having maximum points.
stWrong
  • 143
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How to show the convexity of this set

Is the set, $S=\{\bf x \in \mathbb{R}^n: \sum_{i=1}^{n} \frac{e^{x_i}}{1+e^{x_i}}=1 \}$, a convex set?
Sedi
  • 105
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Solve the linear program

Please help to solve this problem. I am new to this type of problems and any help will be greatly appreciated $$\text{ Minimize } 7x-5y+3z$$ $$\text{ Such that } \ \ \ 0 ≤ x ≤ 6 , -2 ≤ y ≤ 7 , -4 ≤ z ≤ 9$$
Gamini Amerasinghe
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accomodating non-negativity constraint in the dual

Suppose the objective implicitly imposes non-negativity constraint, say, the objective is sum of square roots of the decision variables. Is it necessary to consider the inequality constraints imposing nonnegativity and have dual variable associated…
Hari
  • 239
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Is the following objective function jointly convex?

I have the following optimization problem: $$ \begin{aligned} & \underset{\alpha, \gamma}{\text{minimize}} & & \end{aligned} \frac{1}{2} \|y - \sum\limits_{i=1}^{S}\gamma_{i}\cdot X_{i}\alpha_{i}\|_{2}…
MRashid
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Could anyone give me an example of non-smooth strong convex function?

Could anyone give me an example of non-smooth strong convex function? I cannot figure out one.
Vivian
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Until now, what is the fastest optimization algorithm of non-smooth convex functions

I am wondering if I minimize a non-smooth convex function, which solver should I choose. I think I should choose a fastest one with a big convergence rate. Subgradient descent is always on the textbook. But it is slow. I am seeking a state of the…
Vivian
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Convex Optimization: $\min \left(\sum f(x_i)\right)$ s.t.: $\sum g(x_i) \le C$, where x has no closed-form expression

I am a newbie here and now facing to a hard convex optimization problem, sincerely wish anyone can help me : ) There is a variable vector $\textbf{x}$, and we wish to minimize the sum of $f()$: $$\min~ \left(\sum f(x_i)\right),$$ where $f()$ is a…
Dobby
  • 67
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Quasi convexity and strong duality

Is there a possibility to prove the strong duality (Lagrangian duality) if the primal problem is quasiconvex? Or is there a technique that proves that the primal problem is convex instead of quasiconvex?
Guido
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How to get the minimum and maximum of one convex function?

Condition: $h,f\in \mathbb{C}^{N\times1}, \text{where}f =\hat{f} + e \text{ and } e^H e \leq 1,\ \ \ Q=h^Hff^Hh$. The Lagrangian function of $Q$ is $\mathcal{L} = h^H(\hat{f} + e)(\hat{f} + e)^Hh + \lambda (e^H e - 1) $ which is convex, where the…
begforopt
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The timestep of Forward–backward algorithm

Recently, I try to learn Forward backward splitting algorithm. I find it was proposed in 1988 'Applications of a Splitting Algorithm to Decomposition in Convex Programming and Variational Inequalities'. It is to minimize $$f(x)+g(x)$$ by two steps,…
Vivian
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