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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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Transform SDP Optimization Constraints

A problem of the form was hinted at on an upcoming assignment: $minimize$ $tr(P)$ $s.t.$ $A^TP+PA = -Q$, $P=P^T>0$ It was hinted that the desired solution require the problem be transformed into a form similar to the one shown below and solved…
JDD
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Solution to constrained quadratic optimization problem

Is there a closed form solution to this constrained quadratic optimization problem? $$ \mathrm{argmax}_X\ \mathrm{Tr}(AX)\mathrm{Tr}(BX)\\ 0\preceq X\preceq I $$ Where $A$ and $B$ are hermitian positive matrices and $A\preceq I$, $B\preceq I$. EDIT:…
Ziofil
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Meaning of $\lambda$ on convex fnction definition

What is the meaning (pratical/visual example) and significance of $\lambda$ on the definition of a convex function and why is it set on $[0,1]$?
mremane
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Constrained optimization: problem to establish an inequality

Consider the vector z. The problem is: $\max {z^T x}$, subject to $\ x^T P x \leq 1$, where $P$ is a positive definite symmetric matrix. Show that the optimal value is $\sqrt{z^TP^{-1}z}$. Once this is proof use this to establish the inequaity: $\…
Ptr34543
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dual of $min_x||x-2||$ subject to $||x||_* \leq 1$

I want to find dual problem of $min_x||x-2||$ subject to $||x||_* \leq 1$ Here $x \in R^n$, $2$ denotes vector of $2s$, $||$ is arbitrary norm and $||_*$ is dual norm. So I started with this: $L(x,v) = ||x-2|| + v(||x||_*-1||)$ $g(v) = inf_x(…
Markoff Chainz
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Convexity and number of critical points

Is a smooth, non-constant, continuous function with more than one critical point necessarily not convex? I figure if you have two critical points, you have an inflection point between by Rolle's Theorem. If that inflection point isn't also a…
TurlocTheRed
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How to make this strange constraint convex?

Does anyone know how to make this constraint convex? $$\frac{P_{02}{\gamma_{02}}}{P_{01}\gamma_{02} + P_{U_{2}}\gamma_{22} + 1}> P_{U_{1}}\gamma_{13} + P_{U_{2}}\gamma_{23} + 2\sqrt{\gamma_{13}\gamma_{23}P_{U_{1}}P_{U_{2}}}$$ where only $P_{U_{1}}$…
Peixi Liu
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Prove the inequality relative to $a$ and $b$, both positive numbers

Prove that there are $a, b,$ positive numbers such that: $f(x,y)=x^4+y^4-2(x-y)^2 \geq a(x^2+y^2)-b$ I've tried using the fact that $x^4 + y^4 \geq 2(xy)^2$ and also that $xy\geq -(1/2)(x^2+y^2)$ I've also tried that: $f(x,y)\gt 2(xy)^2 -2(x-y)^2…
Ayoub Boumchich
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Optimum tournament fixture organization

The argentine FA Major league tournament is made up of 26 teams which play in a league system facing each other only once, whether home or away, making up a 25-week-long league. Every team plays either 12 games played away and 13 home or vice-versa…
OrangeWall
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Proving that $(x+y+2z)(2x+4y+z)(x+y+z)$ defines a convex level set

How can I prove that the following function is concave over $x,y,z\ge 1$?$$(x+y+2z)(2x+4y+z)(x+y+z)$$ I tried the following: since $(\prod _{i=1}^n x_i)^ {\frac 1n}$ is concave over $\mathbb R^n_+$, then $$((x+y+2z)(2x+4y+z)(x+y+z))^{\frac 13}$$is…
MasterJ
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Lagrange Multipliers for two constraints, degenerate case

The original post. I was suggested to ask the question here. To optimize $f(x,y,z)$ subject to $g(x,y,z)=h(x,y,z)=0$, we use the Lagrange Multiplier method and solve \begin{equation*} \nabla f=\lambda \nabla g+\mu\nabla h,\quad g=0,\quad…
Haoran Chen
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Re-expressing this as a convex optimization problem.

For some $n$, $\vec{\tau}^{\min},\vec{\tau}^{\max}, \vec{d} \succcurlyeq 0$, and convex,non-decreasing and positive $\Phi(s) : \mathbb{R} \rightarrow \mathbb{R}$, I have the following optimization program: \begin{align*} &\min \sum_{i=1}^n \tau_i…
Television
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Minimum Element With Respect to a Cone

We're asked to find the minimum element of the set S, where $S={\{(0,2), (1,1), (1,2), (4,0)\}}$ , and the minimum element x with respect to K is defined s.t. $y \in S \iff x \preccurlyeq_k y$. The solution is given as (0,2). Why is (0,2) the…
blargen
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What's the difference between linear space and cone?

In the convex optimization domain, What's the difference between linear space and cone? thanks.
YEN_MathStackExchange
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Is this expression concave or convex?

I am trying to check if the dual expression of slide 13 of http://ee364a.stanford.edu/lectures/duality.pdf $- \lambda^TAP^{-1}A^T\lambda - b^T\lambda$ is either convex or concave but am having a hard time. Computing the hessian gives me…
Kong
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