Questions tagged [nonlinear-optimization]

Nonlinear objectives, nonlinear constraints, non-convex objective, non-convex feasible region.

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Minimizing $x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1$

Look at the expression $$f(x_1,x_2,x_3) = x_1^2+x_2^2+x_3^2+x_1x_2+x_2x_3+x_3x_1.$$ The numbers $x_1,x_2,x_3$ are non-negative, and I assume that $x_1+x_2+x_3=3$. This is a sum of squares and "...
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Levenberg's original article "A method for the solution of certain problems in least squares"

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168? It is (...
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Find minimum-area ellipse which encloses two ellipses

I need an efficient algorithm to find the ellipse with the smallest possible area which encloses two given ellipses. The given ellipses are constrained to have coincident centers at the origin but can ...
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What is Young measure?

I read about Young measures from the book, Weak convergence methods for nonlinear partial differential equations by L.C. Evans. He introduces the concept by the following theorem: Theorem. Assume ...
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convert a special case of nonlinear fractional programming into a convex problem

Is it possible to convert a fractional problem (maximization) with objective function equal to the ratio of a concave function and convex function ? This question sound impossible but I have read this ...
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Decomposition of a semi-definite matrix into sums of sparse semi-definite matrices

I'll first provide the background. Let $x\in\mathbb{R}^N$ be decomposed into $n$ non-overlapping blocks of variables $x^{(1)},\ldots,x^{(n)}$. We say that $f:\mathbb{R}^N\rightarrow\mathbb{R}$ is ...
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Solve equation with matrix variable

I want to solve a matrix $\Omega$ from a equation $\sum_k (\Omega + \Theta_k)^{-1} = Q$. The $Q$ and $\Theta, \forall k=1...K$ are known, and are positive definite matrices. $\Omega$ also has to be ...
$v$ is a vector of length $n$, where $v_1 = 1$ and every element $v_i \in [0,1]$ $w = \| v \|_1^1 = \sum_i |v_i| = \sum_i v_i$ $x = \| v \|_2^2 = \sum_i |v_i|^2 = \sum_i v_i^2$ \$y = \| v \|_3^3 = \...