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      Questions tagged [global-optimization]

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      54 views

      How can we analytically solve this max-sum-min problem?

      Let $I$ be a finite set, and $A_{ij},B_{ij},x_i,y_j\ge0$. I want to find the choice of $x_i,y_j$ maximizing $$\sum_{i\in I}\sum_{j\in J}A_{ij}\min\left(x_i,B_{ij}y_j\right)\tag1$$ subject to $$\sum_{i\...
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      1answer
      73 views

      Maximize a Lebesgue integral subject to an equality constraint

      I want to maximize $$\Phi_g(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(w_i(x)p(x)q_j(y)\wedge w_j(y)p(y)q_i(x)\right)\sigma_{ij}(x,y)|g(x)-g(y)|^2$$ over all choices ...
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      41 views

      Optimizing Function over Configuration Space

      Below, I describe a function defined on (a certain subset of) the configuration space of $n$ points on the plane and I want to know how its maximum grows as $n\to\infty$. Before describing the ...
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      4answers
      2k views

      Prove that this expression is greater than 1/2

      Let $0<x < y < 1$ be given. Prove $$4x^{2}+4y^{2}-4xy-4y+1 + \frac{4}{\pi^2}\Big[ \sin^{2}(\pi x)+ \sin^{2}(\pi y) + \sin^{2}[\pi(y-x)] \Big] \geq \frac{1}{2}$$ I have been working on this ...
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      1answer
      520 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
      votes
      1answer
      130 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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      37 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
      61 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 ...
      1
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      1answer
      104 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: $$...
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      0answers
      81 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 ...
      1
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      1answer
      114 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 $1 \leq P \leq N$ ...
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      votes
      2answers
      128 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
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      0answers
      45 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 ...

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