# Questions tagged [nonlinear-optimization]

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

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### Help finding sufficient conditions for unique maximizer to constrained maximization problem

I am working on a paper, and I have run into a constrained maximization problem. I would like to find some sufficient conditions for the maximizer (particularly, p) to be unique. Given my limited ...
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### Optimization problem related to knot placement for parametric interpolation

The problem of knot placement addresses the question of how to choose the parameter intervals $\lbrace[t_i,t_{i+1}]\,|,\, 0=t_0 \leqq t_i\leqq t_{n-1}\leqq t_n=n\rbrace$ in way that renders the ...
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### Backstepping control of second order nonlinear system

$\dot{x_{1}}=x_{2}^2-3sin(x_{1})x_{2}$ $\dot{x_{2}}=x_{1}^3-3x_{2}cos(x_{1})+u^\frac{1}{2}$ Question: Using the backstepping method and Lyapunov function, design the controller $u$ that will make ...
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### Pros and cons of using integer programming alone or combined integer and global optimization?

First, I am not sure if this is the right question to ask in this forum. But I have been looking for answers for a long time and I have been also asking my university's "engineering" professors but I ...
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### Compute which of a finite number of integrals is minimal

Let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $f:E\to[0,\infty)^3$ be a bounded Bochner integrable function on $(E,\mathcal E,\lambda)$ and $p:=\alpha_1f_1+\alpha_2f_2+\alpha_3f_3$ ...
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### What is the jacobian of an image lookup function?

I posted this question on Robotics Stack Exchange (link) but thought it could be relevant here as well. I'm trying to solve a computer vision problem whereby I wish to use Levenberg–Marquardt non-...
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### How to maximum L1 norm problem?

I have met a problem these days. \begin{equation} \underset{\omega}{\max} \quad \Vert \text{diag}(\mathbf{h}^H)\mathbf{G}^H\mathbf{\omega}\Vert_1 \\ s.t.\quad\mathbf{\omega}^H\mathbf{G}\mathbf{G}^H\...
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### A four-variable maximization problem [closed]

We let function \begin{equation} \begin{aligned} f(x_1,~x_2,~x_3,~x_4) ~&=~ \sqrt{(x_1+x_2)(x_1+x_3)(x_1+x_4)} \\ &+ \sqrt{(x_2+x_1)(x_2+x_3)(x_2+x_4)} \\ &+ \sqrt{(x_3+...
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### Show a Poincaré inequality for a Markov kernel and minimize the Poincaré constant

Let $\tilde\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $\tilde Q$ and target distribution $\tilde\mu$ (see definitions ...
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### Is there a closed-form solution for this problem?

Let $A\in\mathbb{R}^{m\times n}$ be a full column rank matrix. Then there exists a left inverse $A^+$ of $A$. Let $w\in \mathbb{R}^n$ be a vector. Is there a closed-form solution for the following ...
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### prove $\log \left[ {\sum\limits_{i = 1}^M {{\varepsilon _i}{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}} } \right]$ is convex

I am having difficulties to prove $\log \left[ {\sum\limits_{i = 1}^M {{\varepsilon _i}{{\left[ {Q\left( {{a_i} + {b_i}\sqrt u } \right)} \right]}^2}} } \right]$ is convex for non-negative a, b,u. ...
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### Is there a multiplier rule for this minimization problem?

Let $(E,\mathcal E)$ be a measurable space, $W\subseteq\left\{w:E\to\mathbb R\mid w\text{ is }\mathcal E\text{-measurable}\right\}$ be a Banach space, $k\in\mathbb N$ and $f:W^k\to[0,\infty)$. I'm ...
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### solving a non-linear Matrix equation

I am working on a problem of optimization of wireless sensor networks in order to achieve the optimal power allocation in the network. I find an equality matrix problem that can be written as ...
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### Understanding an argument involving the Sherrington-Kirkpatrick spin glass model

I have a question regarding an argument in Aizenman, M.; Lebowitz, J. L.; Ruelle, D., Some rigorous results on the Sherrington-Kirkpatrick spin glass model., Commun. Math. Phys. 112, No. 1, 3-20 (...
Let $$G(w):=\sum_{i\in I}w_i\;\;\;\text{for }L^2(\mu)^I.$$ I want to minimize F(w):=\sum_{i\in I}\sum_{j\in I}\int\lambda({\rm d}x)w_i(x)p(x)\int_{\left\{\:pq_j\:>\:0\:\right\}}\lambda({\rm d}y)\...