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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Non-linear programming problem with exponential objective function.
If I have an exponential objective function, but the problem is still convex. What is the best solver to use for my model?
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Geometric interpretation of maximizing utility functions
So this was the problem I was trying to solve, and I got stuck in part (b).
When you total differentiate $U(x,y)=0$, we can see that $Uxdx+Uydy=0$, which means $dy/dx=-Ux/Uy$
However, I am struggling ...
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condition for convex function to have unique minimum value
The question is related to
Convex function with unique critical point is coercive
Assume we have a function $f: \mathbb{R}^{n} \to \mathbb{R}$ and we know that $f$ is just convex, not strictly nor ...
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A Number of Jacobian entries
I'm trying to understand an optimization problem from IPOPT package.
...
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Maximum of $\ell^2$-Norm
For $r,c>0$ put
$$X_{c,r}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r, \, \forall i\in \mathbb{N}: |x_i|<c\}.$$
Then I can show that $\inf_{x \in X_{c, r}} \|x\|_2=0$.
Is it possible to compute
$$...
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Linearization of a binary decision variable [closed]
I tried finding a replacement for the decision variable $X_{ij}$ using linear functions. $X_{ij}$ is the variable and $y_i$ and $y_j$ are integer parameters between $1$ and $8$.
$X_{ij} = 1$ if $y_i \...
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Simplifying large scale nonlinear numerical optimization problem
I want to solve the optimization problem below numerically using GAMS. Given the formulation, I was wondering if there are suitable ways of transforming and or approximating the given function to ...
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Non-linear optimization of a set of initial weights with multiple objectives given a weight constraint
I have an initial set of N weights, W. They sum up to 1.
Now, these weights need to be adjusted such that they fullfill the following constraint:
The aggregated weight of the weights exceeding $\...
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Casino Optimization Problem
A certain game starts with a starting capital C and runs for a number N of rounds. In each round i (with 1 ≤ i ≤ N) the player has the option of not betting anything or betting all of his capital, so ...
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Optimizing a probability for marbles in two buckets
There are 25 red marbles and 25 blue marbles divided between two buckets. You
select a bucket uniformly at random
select a marble from that bucket uniformly at random
Find initial arrangements of ...
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$\inf\limits_{y_i} (\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i) = 0$ if $\lVert \nu_i \rVert_2 \leq \lambda_i$
I am trying to solve the following problem:
$\inf\limits_{y_i} (\lambda_i \lVert y_i \rVert_2 + \nu_i^T y_i) = 0$ if $\lVert \nu_i \rVert_2 \leq \lambda_i$ and $-\infty$ otherwise.
I know that $\...
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How to recover solution of SDP relaxation to maxcut problem given the solution matrix
I have found the solution to an SDP relaxation of the maxcut problem and I have the solution matrix $Y$. I have found that the SDP relaxation was exact because all the eigenvalues of the matrix Y are ...
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Necessary optimality conditions: $\min_{\phi} \sum_\xi\int_{0}^{1} p(a,\xi) T(V_{w(a,\xi)}[\phi]) da$, where $V[\cdot]$ is an evaluation functional
I'm trying to set up a dynamic optimization problem as follows.
Let $\mathcal{W} := [\underline{w},\overline{w}]$, and $w:[0,1]\times\{0,\dots,N\}\to \mathcal{W}$
\begin{align}
\min_{\phi: \mathcal{W}\...
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Does the following inequality hold when $x_i \geq 1 $?
Does the following inequality hold when $x_i \geq 1$ for $i=1,2..k$ and $\sum_{i=1}^k x_i = n$?
$n^2 \sum_{i=1}^k x_i^3 \leq (\sum_{i=1}^k x_i^2)^3$
I observed this while reading an article that this ...
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Quantifying similarity between datasets with non-linear structures using Hilbert Space
Here's a revised version of your Stack Overflow question with some errors corrected and improved clarity:
I have a dataset that represents movements between two locations (origin and destination), ...
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Are KKT conditions still necessary and sufficient for optimality in a nonlinear max problem with pseudo-concave objective?
We all know that KKT conditions are necessary and sufficient conditions for optimality in a convex minimization problem (or a concave maximization problem). Recently, I found that the convexity/...
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Convert a MINLP problem with semi-continuous variables to a problem with continuous variables?
Is there a way do (approximately) convert a nonlinear optimization problem with semi-continuous design variables to a problem with continuous variables? I want to avoid the use of MINLP solvers and ...
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Distribution of Near-Optimal Solutions
For a standard unconstrained minimization problem:
$$\min_{x \in \mathcal{X}} f(x)$$
I am interested in understanding how "big" the set of $\epsilon$-optimal solutions "usually" ...
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How to numerically optimize a function with 'discontinuous' box constraints?
First of all: I am an engineer and no mathematician, therefore please excuse my 'lazy' form or prensenting the problem.
I want to solve a nonlinear and constrained optimization problem:
\begin{align}
...
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How to know when to switch from standard optimization to multi-objective optimization or vice-versa?
To minimize a function $f(X)$ subject to constraints $g(X) \leq0$, there are multiple optimization techniques that can be used (convex optimization if $f$ is convex, differential evolution DE for ...
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Solution for a system of non linear equation
$ x = \phi(x + t)$ is the system of non-linear equation which I need to solve for $x$.
Here $x$, $t$, $\phi(.)$ are vectors all having dimension $n$.
How to solve this system using Newton's method.
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GAUSS-NEWTON algorithm to solve a non-linear system?
I need a parameter to use in Poisson. I want to calculate this parameter using the Gauss-Newton algorithm, starting from a nonlinear system. I want to point out that i am assuming that the ...
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Are there optimization algorithms tailored to (the structure of) a specific function?
As an example, let's consider non-linear least squares (NLLS), which can be solved using Levenberg-Marquardt (LM). LM can be used to solve many different NLLS objective functions, but the LM constants ...
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Solving Lagrange Optimization / System Of Equations / Multivariate Kelly Formula
I am hoping the Stack Exchange community can help me figure out why my Lagrange optimization is outputting a value outside of the constraints.
We have the option of betting on three mutually exclusive ...
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Estimate a vector given pairwise differences
I am not an expert in optimization, so I would appreciate any leads on the following problem, which I have stumbled upon in my research.
Essentially, we are asked to estimate an array $x \in \mathbb{R}...
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SDP relaxation of mixed-integer nonlinear program
I am having trouble understanding the semidefinite programming (SDP) relaxation of a mixed-integer nonlinear program (MINLP) from section 3 of this paper. The optimization problem in MINLP form is
\...
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Optimizing the portfolio in modern portfolio theory
I am trying to understand some aspects of the modern portfolio theory, which has brought me to a point I don't fully understand. I would appreciate any hep/suggestions/references. Lets assume that the ...
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Minimization of a multivariate function where each variable shows sinusoidal dependence with offset
I'm working on minimizing a multivariate function $f(x_1, x_2, ..., x_n)$ and would appreciate some help. The function $f$ has the following properties:
The function's range is bounded: $0 \leq f \...
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Maximizing product of fractions
I want to maximize $f(a,b,c) = \frac{a}{a+1}\cdot\frac{b}{b+1}\cdot\frac{c}{c+1}$, where $a,b,c$ are nonnegative integers and $a+b+c=30$. I began the problem with AM-GM to find an upper bound of 1000 ...
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Applying Cauchy-Schwarz inequality and finding the maximal value of f
I was looking at this example while learning about unconstrained optimization and I don't think I completely understand it.
Here is the example in question: Consider the two-dimensional linear ...
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Primal Interior-Point Methods
It is common that when one solves the nonlinear inequality constrained problem
$$
\min_{x\in\mathbb{R}^{n}}f(x) \\
\text{ such that } c_{i}(x)\geq 0,\,\,\,i=1,2,\dots,m
$$
that one introduces slack ...
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Looking for a Functional Form Satisfying Certain NonLinearites
I'm working with a functional form
$\alpha = F(x,y,t)$ where t represents time and $x,y $ other factors affecting $\alpha$.
I want to find a functional form for F (that also has some parameters) such ...
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Opinion on Barrier/Penalty method for box constraints in Optimization
I had the idea to use an additional term in an numerical optimization problem for a box constraint.
It is somewhat a mix between penalty and barrier function and I am wondering what the drawbacks are ...
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Help on fractional programming
I need help transforming this Mathematical Programming model into a solvable model
$ \min \delta$
$\frac{190(E_j-\sum_ix_{ij})}{P_j-\sum_iQ_ix_{ij}}\leq\delta \ \ \ \ \forall j$
$\sum_j x_{ij} \leq ...
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MILP to LP; is it possible?
There is a machine that can produce $x_t \in [0, \overline x]$ quantities of a good in hour $t \in T = \{1, 2, \ldots, 8760\}$. The production of a unit has linear costs of $k_t \in \mathbb R_+$. The ...
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When the Karush-Kuhn-Tucker conditions fail to apply? [closed]
Consider an optimization problem:
$\max\limits_{\substack{x_1, x_2}} x_1 + x_2$
s.t. $2 \sqrt x_1 + x_2 \leq y$
$x_1, x_2 \geq 0$
In order to solve it, I set up the Lagrangian:
$\mathcal{L}(x_1, x_2) =...
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Rewriting a maximization problem to a minimization one
Suppose that the function $g: \mathbb{R}^n\rightarrow\mathbb{R}$ is convex on $\mathbb{R}^n$ and that $\mathbf{d}\in\mathbb{R}^n$. Is the problem to
$maximize -\sum^n_{j=1}x^2_j$
subject to conditions ...
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Minimization problem $f(x_1,x_2)=x_{1}^2+x_2+e^{x_{1}^2+x_{2}^2}$
"Show that the function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ given by $f(x)=x^2_{1}+x_{2}+e^{x^2_{ 1 }+x^2_{2}}$ has a single point
stationary and that such a point is a global minimizer."
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Choosing optimization procedure for solving non-linear PDE with finite-element method
I'm trying to write a code solving the Dirichlet boundary problem for $p$-Laplacian on an arbitrary planar domain $\Omega$:
$$\begin{cases}
-\Delta_p u = f \text{ on } \Omega, \\
u\big|_{\partial \...
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Implementing gradient optimization methods for non-linear functionals
I'm trying to write a code solving the Dirichlet boundary problem for $p$-Laplacian on an arbitrary planar domain $\Omega$:
$$\begin{cases}
-\Delta_p (u) = f \text{ on } \Omega, \\
u\big|_{\partial \...
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Why do we linearize optimization problems?
I am currently doing research on the calibration of the robots' geometry, which is a standard and well-studied topic. In fact, it can be formulated as a nonlinear non-convex optimization problem:
...
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Question About Fritz John Theorem and Slater Constraint Qualification
Background Information
I am studying constraint qualifications. Here are two theorems leading to my question:
Theorem 1$\space\space\space\space$ [Fritz John Theorem] Suppose that $f, g_1, \dots, g_k$...
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Proof of Second Order Conditions for Maximization Problem with Two Variables and One Inequality Constraint
Background Information
I am studying the theorem of S.O.C.s for a constrained maximization problem. Here is the theorem on the book but lack of proof:
Theorem$\space\space\space\space$ Let $f$, $g_1$,...
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Prove that if the minimizer $\lambda$ of this parabola is feasible in ($*$), we must have $\alpha \in (0, 1/2)$.
The criterion of sufficient decrease (Armijo condition) requires $\lambda \in \mathbb{R}$ such that:
\begin{equation}
\varphi(\lambda) = f(x + \lambda d) < f(x) + \alpha \lambda \nabla^T f(x) d = \...
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Constrain set of a constrained optimization problem
In the lecture notes I have, the constrain set in the equality case is defined as
$S \cap \{x\in \mathbb{R}^n : g(x)=0\}$ where $S \subseteq \mathbb{R}^n$ open.
The inequality and equality-inequality ...
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Weighted least squares closed form solution
Warning - it's been awhile since I've dived into matrix+optimization math, so bear with me.
I'm trying to determine the center of rotation of a single 3D link (think a rod attached to a ball joint) ...
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About Implicit Function Theorem and Lagrange Multipliers
I am studying the meaning of the multiplier in Lagrange Multiplier Method and got the following question. I tried it myself, but I am not sure if it is correct. I would really appreciate it if someone ...
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The Intuition of Non Degenerate Constraint Qualification and Its Application When The Number of Binding Constraints Is More Than That of Variables
I am studying constrained optimization, and I am a bit confused by the Non Degenerate Constraint Qualification (NDCQ), its intuition and application. First, I would like to put the theorem of ...
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Find parameters of a gaussian transformation
I have a system of equations where the relationship between input and output is derived from a pixel lattice:
\begin{equation}
x_i(k+1) = \sum_j \alpha e^{ \frac{\left(dr_{ji}^2 + dc_{ji}^2 \right)}{2 ...
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Particle Swarm: Restart a stalled particle near the "global best" or the particle's "local best" position?
I am working on an implementation of Particle Swarm that intelligently restarts a particle when its velocity reaches zero, so the particle can find a new starting point and continue its search. This ...
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