# Questions tagged [nonlinear-optimization]

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

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### Is this gradient-descent-like algorithm which creates sparity

Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a class $C^{1,1}$-function, $\lambda\geq 0$ be a ''learning rate'', $\lambda\geq 0$ be some "sparity generating parameter", $T\in \mathbb{N}$ be ...
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### Lagrange Optimization w.r.t. vector + equality constraint

I am currently facing the following optimization problem: $f(v) = r_f + v'*\mu + v'*diag(\Sigma)*1/2 -v'*\Sigma * v*1/2+(1-\gamma)*v'*\Sigma*v*1/2$ s.t. $1'*v=1$ (sum of the vector elements = 1) ...
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### Transformation of an unconstrained binary quadratic optimization problem into a constrained binary linear programming problem

I know that a constrained linear optimization problem can be transformed into an unconstrained binary quadratic optimization problem (UBQP). Does anyone know if the inverse result is solved in the ...
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### What's "Arrow-Hurwicz method" for solving saddle point optimization problems?

I have seen some papers on convex-concave optimization citing the "Arrow-Hurwicz method" from the paper  in different ways. However, since I cannot find a pdf version of this paper and ...
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### Relaxations for the spectral norm maximization problem

Optimizing the spectral norm of some positive semidefinite matrix $A(x) \in S^{n}$, w.r.t. a list of variables $x \in \mathbb{R}^d$ and semidefinite constraints is, in general, a nonconvex problem (...
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### Eigenvectors of a tensor in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$

I want to find the critical point of tensor $f=a_0b_0c_0 + a_1b_1c_1$ in $\mathbb{C}^2 \otimes \mathbb{C}^2 \otimes \mathbb{C}^2$, and I followed this construction: First, I take the following partial ...
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### Are such assumptions of functions similar to strong convexity reasonable in convex optimization?

For $\mu$-strongly convex function $f:\mathbb{R}^d\to\mathbb{R}$, the following property holds: for any given $x,y\in\mathbb{R}^d$, we have $$(\nabla f(x) - \nabla f(y))^\top(x-y) \ge \mu \|x-y\|^2.$$...
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### Condition under which the Clarke's subdifferential is locally Lipschitzian

Given a locally Lipschitz continuous function $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$ and closed set $$\Omega =\left\lbrace x \in\mathbb{R}^n \ |\ f(x) \leq 0 \right\rbrace$$ such that f is semi-...
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### Do Pareto critical points of a multicriteria optimization problem form an attractor of the dynamical system induced by a descent algorithm?

Let $d\in\mathbb N$, $k\in\mathbb N$ and $f:\mathbb R^d\to\mathbb R^k$ be differentiable. Say that $v\in\mathbb R^d$ is a descent direction at $x\in\mathbb R^d$ if ${\rm D}f(x)v<0$ (component-wise) ...
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### Minimizing the largest eigenvalue of matrix product

Let $A\in\mathbb{C}^{m\times n}$, $B\in\mathbb{C}^{n\times k}$, $C\in\mathbb{C}^{k\times m}$ be given complex matrices. The objective of the optimization problem is \begin{equation} \mathop {\arg \min ...
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### Example of a differentiable function optimization where derivative free methods are used

While preparing a workshop on the derivative free methods, and fminsearch in MATLAB, I found an example function where fminsearch converges better and in less iterations than fmincon with calculated ...
Given real numbers $y_i's$, consider the following convex optimization problem: $$\min_{x_i's} \sum_{i=1}^N(y_i-x_i)^2 + \lambda\sum_{i=1}^{N-1}|x_{i+1}-x_{i}|.$$ The paper A Direct Algorithm for 1D ...