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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.

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. Usually, non-linear optimization problems are much harder to solve than linear ones.

2947 questions
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Determining the minimum value

Function $W(t,x)$ is defined as \begin{equation} W(t,x)=\sum_{i}\alpha_i e^{-\beta_i(t-x)}, \end{equation} where $\alpha_i$ is real and $\beta_i$ is real and positive, Then $\Psi$ is defined as \begin{equation} \Psi =\int_{a}^{b}\Big{(}h(t)…
user157127
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Identifying saddle points of a constrained nonlinear function with three variables

I know that if the Hessian matrix of a multivariable function at a given stationary point has both positive and negative eigenvalues then that stationary point must be a saddle point. Does the same hold with multivariable optimization problems with…
Ryan G
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Unconstrained Nonlinear Nonconvex Optimization: LBFGS vs. Interior Point Methods?

I'm finding the literature on interior point methods somewhat inaccessible but I've found papers benchmarking different interior point methods for unconstrained nonlinear Nonconvex optimization. I can't find a comparison between interior point…
Praxeolitic
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Is it possible to convert into linear programming problem

I have a problem of the form $$\sup_{x\in\Bbb{C}^n}\left\{\frac{\|Ax\|_\infty}{\|Bx\|_\infty}\right\}$$ where $A$, $B$ are matrices with different number of rows and $x$ is an $n$ dimensional vector. Is there a way to find a tight bound to the…
ARM
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What is inexact steepest descent method

Is there anybody knowing what is the inexact steepest descent method for solving non-linear optimization problems? Any reference or formal definition available online? I was asked by someone, but didn't find the best answer. I think it should be an…
user42226
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fractional optimization problem with equality constraints

I have the following optimization problem \begin{equation} \begin{aligned} \max_{\mathbf{x}} & \ |d-\sum_{n=1}^{N}\frac{c_n}{f_n+x_n} |^2 \\ \quad \text{subject to} \quad & \sum_{n=1}^{N} \frac{|a_n|^2 Re(x_n)}{|f_n+x_n|^2} =…
Isoto
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Minimizing a single-variable quadratic function with quadratic constraint on the same variable

I am reading a paper (see Algorithm 3.1) that contains the sub-problem: Find $\tilde{\tau}$ so that $\tilde{s} = \tilde{s}_j + \tilde{\tau} \tilde{p}_j$ minimizes $$ \tilde{\phi}(\tilde{s}) = \tilde{g}^T\tilde{s} + \frac{1}{2}\tilde{s}^T\tilde{B} \;…
Olumide
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Proof of uniqueness of point in closed, convex set with minimum distance from another point not in the set

I am reading a proof of Theorem 2.4.1 in "Nonlinear Programming" by Bazaraa, Sherali, and Shetty, and I am wondering what the reasoning is on a step of the proof. I am almost certain it is something very simple I am blanking on. The Theorem…
user316720
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How to derive the KKT-conditions?

The following non-linear optimalisation problem: \begin{aligned} & \quad \min x_1 \\ & \text { } x_1+x_2=0 \\ & \quad\left(x_1^2+x_2^2-4\right)^2=0 \end{aligned} The task is to find the KKT-conditions. The solution says that: \begin{aligned} &…
Tim
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Maximum of $\textbf a^{ T} \textbf x$ over $ B[0,1]$ is $\textbf x^*=\frac{\textbf a}{\lVert \textbf a \rVert}$

Let $\textbf a \in \mathbb R^n$ be a nonzero vector.Show that maximum of $\textbf a^{ T} \textbf x$ over $ B[0,1]=\{\textbf x \in \mathbb R^n: \lVert \textbf x \rVert \le 1 \}$ is attained at $\textbf x^*=\frac{\textbf a}{\lVert \textbf a \rVert}$…
user1040538
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Is this coercive function?

Given a function $f(a) = a^4 +a^3 +2a^2 +a+1$ . How can I show that it is a coercive function? I am having a hard time understanding how to calculate the norm of $f$ in order to show that f is coercive. Can someone help me? Thanks!
Blake
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A non-linear optimization problem with constraints

We have two constraints: $$ \left[p_1+ q_1e^{\gamma s_1 (1-\alpha_{11}^{*})}\right] \left[p_1+ q_1e^{-\gamma s_1 \alpha_{11}(n_1-1)}\right] \left[p_2+ q_2e^{-\gamma n_2s_2 \alpha_{12}}\right] = 1 $$ and $$ n_2\alpha_{12}+(n_1-1)\alpha_{11} +…
Ilonpilaaja
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Optimize area under curve of $f(t)$ under constrains $f(t)$ obtains maximum at $t_{max}$ and $f(t) \leq g(t)$

Let $g$ be defined by $$ g(t) = a + b\, t + c\, \mathrm{exp}(-kt), $$ where $a > 0, b < 0, c < 0$ and $0 < k < 1$ (for my use case I have $k = 0.05$). $a$, $b$, and $c$ are chosen such that $0 \leq g(t)$ for $0 \leq t \leq t_{end}$. A specific…
Noud
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Is a strictly global minimum a unique global minimum?

A strictly global minimum of a nonlinear continuous twice differentiable function seems to imply that this minimum is its unique global minimum. How this can be proved or disproved?
Hass Saidane
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Optimality Conditions for functions with Lipschitzian Gradients

I am looking for sufficient optimality conditions for problems of the type $$ \min_{x} \lbrace f(x) | x \in X \rbrace, $$ where $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable with Lipschitzian gradient, i.e. $f \in C^{1,1}$ and $X$ is a closed…
Marita
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