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      Questions tagged [global-optimization]

      The tag has no usage guidance.

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      0answers
      111 views

      KKT for dual of a quadratically constrained linear program

      Let $ \mathcal{P}$ be a linear program with quadratic constraints, $$ \eqalign{ \mathcal{P}: & \min_{x,t} -t \\ & \text{s.t.} \\ {\color{orange}{\mu_k}}: & a^H_k x + a^T_k x^{*} - x^H ...
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      1answer
      490 views

      Illustrating that universal optimality is stronger than sphere packing

      I'm a physicist interested in the conformal bootstrap, one version of which was recently shown to have many similarities to the problem of sphere packing. Sphere packing in $\mathbf{R}^d$ has been ...
      5
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      1answer
      122 views

      An effective way for the minimization of $\left\|ABA^{-1}-C\right\|$

      Supposing I have complex square matrices $B_i$ and $C_i$ ($i = 1,\dots,N$) of dimension $4 \times 4$. Is there an effective algorithm for solving the following problem? $$\begin{align} A=&\...
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      19 views

      Can the partial concavity of the following decomposable objective function be used for optimization?

      The problem I am trying to solve is the following: $$\begin{array}{ll} \min & f(x)+g(y) \\ \mathrm{s.t.} & y\ge x\ge 0,\\ \ & p\le ax+by\le q, \end{array}$$ where $a,b,p,q$ are ...
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      36 views

      Stochastic process for minimize a mean

      I have the next problem: consider an inventory process $\{X_{k}\}$ such that $X_{k+1}=X_k+A_k-\xi$, $X_0=25$, where $A_k$ is the number of items of items produced at the $k$th-month and $\xi$ is the ...
      5
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      0answers
      60 views

      On a maximum of a determinant with dependent variables

      Let $x_1,\ldots,x_n\in [-1,1]^n$ and define the function $$f(x_1,\ldots,x_n):= \prod_{i=1}^n\prod_{j=i}^n\left(1-\prod_{k=i}^j x_k\right).$$ This is a positive function, and actually coincides with ...
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      1answer
      96 views

      Minimizing sum of ratio of linear functions (Sum of Linear Ratios Problem)

      Given constants $c_i \in \mathbb{R}$ and $d_i \in \mathbb{R}$ and variables $x_i \in \mathbb{R}$, where $c_i > 0, d_i > 0, x_i > 0$ can we easily solve the following optimization problem: $$...
      1
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      0answers
      78 views

      When are quadratic integer programs “easy to solve”?

      Let $N_i=\{0,1,\dots,\bar{n}_i\}$ and define $N=N_1\times \dots \times N_I$. I want to maximize $f$ on $N$. $f$ has the following form $$ f(n) = \sum_i A_i n_i -\sum_i \sum_{j\neq i} B_{ij} (n_i-n_j)^...
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      32 views

      symmetry preservation and symmetry breaking in optimization algorithms

      Let $M$ be a Riemannian manifold, and $G$ be a group of isometries on $M$. If we have a $G$-invariant functional $F: M \rightarrow \mathbb{R}$, i.e. $F(g \cdot x) = F(x)$ for all $x \in M$ and $g \in ...
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      1answer
      102 views

      Finding $P$ points among $N$ to approximate a probability density function?

      Let $f$ be a probability density function (positive such that $\int_{\mathbb{R}} f(x) \mathrm{d} x = 1$) and $X_0 = \{x_n\}_{1\leq n \leq N}$ be $N$ given real points. We also fix $1 \leq P \leq N$ ...
      2
      votes
      1answer
      92 views

      Looking for a very particular kind of non-convex functions

      I want some examples (hopefully parametric families!) of non-convex functions which satisfy the following properties simultaneously, It should be at least twice differentiable. It should have a ...
      1
      vote
      0answers
      41 views

      Minimum Preserving Transformations [closed]

      If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y ...
      0
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      1answer
      202 views

      Properties of the argmin function (continuity, differentiability..) [closed]

      Given vectors $x_1,\ldots,x_N \in \mathbb{R}^d$ and a function, say $\psi \colon \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}$, one is interested in the properties of the function $$\Phi\colon (x_1,...
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      2answers
      259 views

      Maximizing a function that is sum of gaussians

      Let $\mathbf{x}_1,\dots,\mathbf{x}_n$ be given $n$ vectors in $\mathbb{R}^d$. Define the function \begin{align} \mathcal{K}(\mathbf{x},\mathbf{y})= \alpha\exp(-\frac{||\mathbf{x}-\mathbf{y}||^2}{2\...
      4
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      2answers
      135 views

      Optimization problem on trace with both the positive semi definite and non positive semidefinite matrix

      Given two $N \times N$ symmetric matrices $A, B$, where $A$ is positive semidefinite while $B$ is not positive semidefinite. I am interested in solving unitary constrained trace maximization problem: ...

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