Mathematics Stack Exchange
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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.

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Non-linear optimization: Levenberg-Marquardt gives different results using different forms of the same function

Background: I'm using non-linear least squares to solve indoor trilateration problem using BLE (bluetooth low energy) beacons. My starting point is this: https://nrr.mit.edu/sites/default/files/documents/Lab_11_Localization.html B part of the…
bartfer
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Formulate the Langrangian function of a non-linear optimization problem and solve it for $y\geq0$

Consider the (non-linear) optimization problem ($P$) $$max \quad3x_1 + 4x_2$$ $$s.t. \quad x_1^2 + x_2^2 \leq 25$$ $$ \quad x_1,x_2 \geq 0$$ Formulate the Lagrangian function $\varTheta(y)$ and solve it for fixed $y>0$ This is the function that I…
dreamer
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Linearizing a program with multinomial logit in the objective

I have a nonlinear problem as follows: $$ min \sum_{k=1}^{K}|y_k - \sum_{i=1}^{N} \frac{e^{x_{k}^{i}}}{\sum_{j=1}^{K} e^{x^{i}_{j}}}| \\ st \quad x^i_{j} \ge 0$$ Essentially, there are $k$ buckets with a desired value of $y_k$ for each. There are…
Alex
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Partial alternating minimization for solving generalized consensus optimization

I'm trying to solve to the following generalized consensus optimization problem, which is a particular kind of nonlinear nonconvex problem: $$\begin{equation} \tag{P} X^*=\underset{x_i \in \mathcal{X}_i,~ \forall i \in \{0,1,\dots,N\} \\ y \in…
duburcqa
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Standard Wolfe condition for matrices

I have an optimization problem that I want to solve with conjugate gradient descent method. Definition is as follows: $$\operatorname{argmin} f(X)$$ Where $$f(X)=\sum_{s=1}^{T} \left \| F_sX(:,s)-Y(:,s)))\right \|_{2}$$ I want to obtain the…
strahd
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Minimize the sum of exponentially increasing values with a product constraint

I would like to minimize $$\sum^N_i \sqrt[2^i]{x_i} = x_0 + \sqrt{x_1} + \sqrt[4]{x_2} + \cdots$$ subject to the constraints $$1 = \prod^N_i x_i = x_0x_1x_2\cdots $$ and $$x_i \in (0, 1]$$ Is this even a well defined problem? I am a bit out of my…
Daniel King
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Why only Nelder-Mead simplex method?

For constructing simplex, we can use any combination of contraction, shrink, reflection and expansion. So why we use the particular order that is being described in Nelder-Mead simplex method?
ANJAN SAMANTA
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Non-linear least squares, problem with shifts

There is a set of coordinates $P=\{P_i\}$. $P_i=[x_i,y_i]$ and a set of coordinates $Q=\{Q_i\}$, $Q_i=[X_i, Y_i]$, where $Q_i$ coordinates are given by the following non-linear functions $$X = f (a_1, a_2, a_3, ..., a_7)$$ $$Y = g (a_1, a_2, a_3,…
justik
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Solution to Fermat Point optimization problem.

There are 3 points in a plane $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$. I am trying to find a solution to the unconstrained minimization problem which is the point $(a,b)$ of least distance between the 3. $\min F(a,b) = \sum_{i=1}^3…
CharlesBouwer
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A non-linear maximisation

We know that $x+y=3$ where x and y are positive real numbers. How can one find the maximum value of $x^2y$? Is it $4,3\sqrt{2}, 9/4$ or $2$?
Basil R
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Find a direction that minimizes the function

Let's say I have a function $f:\mathbb{R^n} \rightarrow \mathbb{R}, f \in C^2 $ and $x^k \in \mathbb{R^n}$ such that $\nabla f(x^k) = 0 $, but $\nabla ^2f(x^k) $ is not positive semi-definite, can I find a direction $d$ such that a point $x^{k+1} =…
Rael
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Significance of Whitney's theorem

Consider the nonlinear optimization problem below: $$p^* := \min f(x)$$ $$\textrm{s.t. }\ g_i (x) \leq 0,\ \forall i = 1,...,m$$ $$h_j (x) = 0,\ \forall j = 1,..., p$$ $$x \in D$$ where $f, g_i, h_j : D \rightarrow R, i = 1,..., m, j = 1,...,p$ are…
Teodorism
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how to choose Marquardt parameter when the variable have different scale

I knew the the Marquardt parameter could switch the different edition between Gauss-Newton and steepest descent method , It add a lambda at the diagonal of Hessian matrix but what if the jacobi of different variable have different order of…
Mr.Guo
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Inequality constrained problem with the number of binding constraints equal to the number of choice variables

I am trying to solve the problem below using basically Theorem 2 that is presented in these notes. It describes the usual method of solving the problem of static optimization when you have equality and inequality constraints. The basic idea is to…
DanielTheRocketMan
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Efficient conjugate gradient method against BFGS for nonlinear optimization

I am trying to optimize an unconstrained non-convex optimization problem. I know that, in general, BFGS takes fewer iterations than conjugate gradient method but consumes more memory (and hence sometimes more computational time). However, I…
user402940
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