# Questions tagged [nonlinear-optimization]

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

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### Inequality Involving Concave Monotonic Function

Assume that $f: \mathbb{R} \to \mathbb{R}_+$ is a concave, non-decreasing and positive function. Let $\mathbb{X}$ be a finite set consisting of $0\leq x_1 \leq x_2 \leq x_3 \leq \ldots \leq x_n$. ...
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### When can a point be reconstructed from relative angle measurements?

Given a set of points $p_1,\dots,p_n$ in $\mathbb{R}^d$ and a target point $x\in\mathbb{R}^d$, I measure all the angles between all pairs of points and the target point. In other words, I have the ...
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### Does this maximisation problem admit a finite upper bound?

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...
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### How to formulate piecewise quadratic function optimization without introducing binary variables?

I have a problem with logical constraints (either-or constraints). I know that it can be solved by either big-M or complementary formulations. However, i do not want to convert it into mixed-integer ...
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### Minimizing the Spectral Norm of the Hadamard Product of a Quadratic Form Using CVX

I am trying to use CVX to minimize the spectral norm of the Hadamard product of two matrices, one of which is in quadratic form. Specifically, I am trying to minimize $\|{\bf A} \odot {\bf XX}^H\|_2$, ...
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### How to prove the convergence of Gechberg-Saxton algorithm?

I just have a problem that Gerchberg-Saxton algortihm is no worse than the previous iteration but not sure whether it is convergent.
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### Equivalence of minimizing trace and determinant over matrix quadratic form in multivariate regression

Consider the multivariate regression model $$Y = XB + E$$ where $Y$ is $n \times p$ and corresponds to the dependent variables, $X$ is $n \times k$ and corresponds to the independent variables, $B$ is ...
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### Relationship of optimal solutions between the total function and the sub function

This is an unconstrained convex optimization problem. Let $\mathcal{N}=\left\{1,\ldots,n\right\}$, $2\leq n<\infty$. Suppose there are many strongly convex functions $f_i(x)$, where \$x\in\mathbb{R}^...
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I am interested in the value function of a quadratic program of the form $$v(y)=\min_x \frac{1}{2} x^\top Q(y) x,$$ subject to a linear equality constraint $$E(y)x=d(y),$$ and a linear inequality ...