# Questions tagged [nonlinear-optimization]

A non-linear optimization problem includes an objective function (to be minimized or maximized) and some number of equality and/or inequality constraints where the objective or some of the constraints are non-linear. Use this tag for questions related to the theory of solving such problems or for trying to solve particular problems.

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### Uniqueness in minimisation of Lp-Norm using Gradient Descent

I am trying to estimate the path of a random described by the following SSM ...
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### Gradient of an objective function containg coupled odes

I have position values of the form $\mathbf{X} = \left[x \; \;y \right]^{T}$. Frenet-Serret model for 2D would consist of following equations: $$\frac{d{\mathbf{X}}_{model}}{dt} = V(t){\mathbf{T}}$$ ...
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### Constrained optimization of quadratic min function

Say I have an objective function $f(\mathbf{x}) = \min ( x_1^2, a_1 ) + \min ( x_2^2, a_2 )$ that I want to maximise subject to constrains $x_1 \geq b_1$ and $x_2 \geq b_2$. This obviously isn't ...
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### Split Bregman Algorithm for L1 optimization

I know that split bregman algorithm can be used for $L1$ norm optimization problem. In literature I have seen solving the problem of $x =: \underset{x}{\text{argmin }}\frac{1}{2}||y-Ax||^2+||x||_1$ ...
1 vote
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### How is the Wilson-Han-Powell SQP algorithm applied?

Say for example we need to minimize $x_2$ subject to $x_1^2+x_2^2-1=0$ starting at $x_1=x_2=1/2$ and using $B=\nabla^2[x_2+\lambda(x_1^2+x_2^2-1)]$ with $\lambda=1$. Now, the WHP-SQP algorithm goes ...
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### Proof that markov chain equilibrium using Farkas' lemma

Given a transition matrix for markov chain $P \in \mathbb R^{dxd}$ such that $$P_{i,j} \geq 0,\quad 1 \leq (i,j) \leq d, \quad \sum_{j=1 \in d }P_{i,j}$$ and $i=1,....,d$. Let $x_{0}$ be ...
1 vote
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### Comparison of Wolfe conditions to other "weaker" conditions or facts about optimization techniques for conceptual understanding

So in the book, it states that the first wolfe condition is the following, $\begin{equation}p^Tg_k\leq-\eta_0|||p|||g_k||\end{equation}$, where $g_k=\nabla F(x_k)$. Here it states that this is a ...
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### How can we solve this geometry problem using Lagrange Multipliers? It is must to use the given formula of area.

The problem is belongs to Mathematical Modeling with Excel book.
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### Uniqueness of nonlinear least squares problem

Consider the problem $$\sum_{i=1}^m g_i^2(x) \to \min$$ with positive $\partial g_i/\partial x_k$, $x \in \mathbb R^n$, $m>n$. Can this problem have non-unique local minimum? What if we solve this ...
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### Gradient descent on a nonconvex nonsmooth function

Suppose we have a nonconvex function $f: \mathbb{R}^c \to \mathbb{R}$ that could potentially be nondifferentiable at some points (but still is continuous - e.g, sum of absolute value functions with ...
1 vote
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### Pontryagin Maximum Principle with terminal and initial conditions

Consider a control problem with Lagragian $L(t,x,u)$ (where $u$ is the control, $x \in \mathbb{R}^d$ the state) and dynamics $\dot{x}=f(x,u,t)$. I have mostly seen problems in which the dynamical ...
1 vote
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### Problem with second order optimality in KKT

I'm solving an optimization problem with KKT method and I've found set of solutions for the problem that meat KKT conditions but now I'm trying to use second order optimality condition for my answer ...
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### Optimization Algorithm for Continuous Objective but Binary Nonlinear Constraints

Is there an derivative-free local optimization algorithm for a continuous function with nonlinear constraints, where the constraints are binary? In other words: $$\max_{x \in \mathbb R^n} f(x)$$ ...
1 vote
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### Paradox definition of a function and its domain

I am currently reading and working through a chapter of "Convex analysis and nonlinear optimizations" by Borwein and Lewis (Chapter 3). At the beginning of the chapter I came across this ...
1 vote
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### Is the $\operatorname{argmin}$ of a uniform strongly convex function continuous?

Let $f:\mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ be a continuous function, and assume that $f(x,y)$ is $\mu$-strongly convex in $x$, for some $\mu>0$ and for any $y \in \mathbb{R}^m$...
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### Solving a nonlinear optimization problem

I'm trying to solve the following problem: Let $\varphi_i: \mathbb{R}^n \to \mathbb{R}$ be continuously differentiable and $\varphi_i( {\bf 0} ) = 0$ for all $i=1,\dots,r$. Assume that $x^*$ is a ...
1 vote
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### One-dimensional heat conduction

I was reading a paper about one-dimensional heat conduction problem and I get stuck in one expression that I couldn't understand how to calculate. First, they define the heat conduction problem as ...
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### Transforming an unconstrained optimization problem into a two-step constrained optimization problems

I have the following problem: Problem 1: $\arg\min_{x,\vec{y}} g\left(\frac{f(x, \vec{y})}{\|\vec{y}\|}\right)\|\vec{y}\|$ where $g$ is monotonically increasing, $f$ is convex and $\|\cdot\|$ is some ...
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### How come nonlinear optimization problems need careful choices of initial parameters but neural networks appear to not have this issue?

When I run some nonlinear optimization code - I often encounter people saying that there is no global nonlinear optimization code that is guaranteed to reach a global maxima. Instead it is recommended ...
I am reading Non-negative Matrix algorithm using KL-divergence as metric. The KL-divergence is known as $D(P,Q)=\sum_i P(i)log\frac{P(i)}{Q(i)}$ for discrete distribution. However, the KL-divergence ...
Suppose we have a number of noisy space-phase data, i.e., $(x_i,y_i,\phi_i)$, and we know they are subject to a 2D linear phase pattern. That is to say, these data can be fitted using a 3-parameter ...