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 ...

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nonlinear KKT analysis with bounded variables

By the help of lagrangian multipliers, I am solving a nonlinear problem with 6 variables using KKT analysis. At the beginning, I do not consider any upper-bound for the variables to see whether the ...
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47 views

Maximising sum of sine/cosine functions

I have got a problem and I would appreciate if one could help. I have to maximise following function that is the sum of sine/cosine functions: $$ f(x,y)=a_1 \cos(x) +b_1 \sin(x)+ a_2 \cos(y) +b_2 \...
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22 views

Nested symsum and symprod.

I am trying to use fmincon to find the values of $w_j$ which minimize the following monster (wherein t is a known constant): $$\...
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12 views

Nonlinear Multivariate Regression

Assuming I know exactly my forward model, which is represented by $n$ non-linear functions, or some probability models: $\vec{R}=f(x,y,z)$ , $f:\mathbb{R}^3\to\mathbb{R}^n$ Where each item in $R_i$ ...
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35 views

Solving a nonlinear equation $\sum_{z=0}^{s} \frac{(\lambda(l-x))^z}{z!} e^{-\lambda(l-x)}=p$

I would appreciate it if someone helps me with solving the following equation. Suppose $\lambda,l \in R^+$, $p\in[0,1]$, and $s\in N_{0}$. How can we find an $x\in [0,l]$, which satisfies the ...
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16 views

Finding solution to Calculus of Variation of linear functional whose domain consists of vector valued function

Problem Statement: Find $x^*$ such that it solves the optimization problem $$\max_{x \in \Omega} \quad f(x) = e_i^TAx$$ $$ \Omega = \{x: t \to \Delta^{n}|x \in C^1, x(0) = x_o\}$$ Where $\Delta^...
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69 views

Find maximum value of a function [closed]

$a$, $b$, and $c$ are real numbers, and $a+b+c=0$ and $a^2+b^2+c^2=2$. I need help finding the maximum value of: $$\big|a^2b^2(a-b)+b^2c^2(b-c)+c^2a^2(c-a)\big|$$ To be honest, I don't know where ...
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1answer
17 views

Multi Valued Model

Suppose we have a set of observations $\{(x_1,y_1),...,(x_n,y_n)\}$ and a function $f$ that provides an optimal (in some sense) prediction of an output given an input. Also suppose that for some $i$ ...
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87 views

finite polynomials satisfy $|f(x)|\le 2^x$

This is a problem from TsingHua University math competition for high school students. Prove there exists only finite number of polynomials $f\in \mathbb{Z}[x]$ such that for any $x\in \mathbb{N}$ ,...
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39 views

Convex optimization with $\ell_0$ “norm”

I have an optimization problem of the form $$\begin{align*}\text{minimize }\;&f(x)\\ \text{subject to }\;&||x||_0 \le t,\end{align*}$$ where $t$ is a given constant and $f:\mathbb{R}^d \to \...
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1answer
34 views

What is the easiest way to optimize the weighted sum of L2 norms?

I have the following cost function (solving for $M$ - the $x_i$s are known): minimize $\sum_i\sum_j(w_{ij} \cdot (x_i-x_j)^T\cdot M\cdot(x_i-x_j))$ ($w_{ij} \in [-1,1] $) subject to: $M \succeq 0$ (...
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1answer
41 views

Linear programming with a product term in the objective function

The title might sound a little weird. I actually want to ask if this problem can be solved as a LP. And if so, how to convert the product term? set $P=\{1,2,3,\ldots,n\}$ for index $i$. Variables $...
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2answers
68 views

Find the smallest $\alpha$ such that, for all $x,y,z$, $\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1$.

Find the smallest $\alpha\in\mathbb{R}$ such that, for all $x,y,z\in\mathbb{R}$, the following inequality holds $$\alpha\,\left(x^2-x+1\right)\left(y^2-y+1\right)\left(z^2-z+1\right)\ge(xyz)^2+|xyz|+1\...
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53 views

Optimization with box constraints - via nonlinear function

I have the following convex optimization problem over $\mathbb{R}^n$ with box constraints: $$\begin{align}\text{minimize }&\;f(x)\\ \text{subject to }&\;x \in [-1,1]^n\end{align}$$ I can see ...
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39 views

Is there a way to measure how (non)convex a function is, maybe analogous to condition number?

Consider the functions $f(x) = \sin x$ and $g(x) = (x+1)^2 (x-1)^2$. We know that $f$ has an infinite number of local minimizers and is nonconvex on a non-compact subset of $R$. We know that $g$ has ...
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1answer
30 views

Smallest Eigen value of Matrix + other matrix less than greatest value eigen value of Matrix

I've been working on this problem for a little bit and I'm not sure if it can be proven with the given information. Any help would be greatly appreciated to either confirm or deny my suspicion. ...
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17 views

Find optimum diagonal matrix $D$ to maximize $ADB$ above a threshold $\gamma$

I have a problem to find the optimum diagonal matrix $D$, which would maximizes the number of elements in $ADB$ which are above a certain positive number $\gamma$. In other words, the problem is ...
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21 views

Introduction to morse theory with applications to optimization

I am wondering if there are any easy-to-read introduction materials on morse theory (especially with applications to nonconvex optimization) for people with non-math background.
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14 views

Finding Dual of non-standart programming problem

I am working in optimization field. My programming problem is not of the standart form, however it is convex. Objective is nonlinear but concave (log of product). I do maximization. Constaints: ...
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23 views

Does converting an inequality constraint to an equality one have any major impact on an optimization solver?

In an optimization problem, I have an inequality constraint, say $\begin{array}{c} {\min\limits_x~} c(x)\\ {s.t.~}g(x)\le 0 \end{array}$ The function $g(x)$ in general is unknown. So, numerical ...
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24 views

Sequential versus simultaneous optimization of multivariate problems

Suppose we have the bivariate function $f(x,y)$. I want to solve the following problem: \begin{equation} \min\limits_{(x,y)} \; \; f(x,y) \end{equation} I want to prove theoretically that ...
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73 views

Can this optimization problem be solved?

I am working on an optimization problem but I am not sure if the problem can be formulated as an integer programming problem. Assume the cost minimization problem for a set of subscribers and ...
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56 views

Is Frank Wolfe a descent algorithm?

A colleague was explaining to me that the Frank-Wolfe algorithm is a descent algorithm (i.e. its objective value decreases monotonically at each iteration). However, when I tried simulating it, my ...
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16 views

Sum of convex and decreasing function

I have a sum of decreasing function and a convex function over some domain. Can I say that the sum is also a convex function (i.e. there exists a unique minimum)?
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36 views

Maximizing the sum of the squares of numbers whose sum is constant

I wonder how one goes about to find the maximum of $\sum v_i^2$, the $v_i$'s being positive integers whose sum $\sum_i v_i$ is fixed.
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1answer
19 views

Why does Frobenius norm make BFGS scale-invariant?

On slide 11 here it is claimed that the weighted Frobenius norm leads to a scale-invariant optimization method. Similar claims about this norm can be found throughout the literature see 1,2,3. In ...
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1answer
30 views

Hessian matrix of the mahalanobis distance wrt the Cholesky decomposition of a covariance matrix

I'm stuck with the following problem: I have to compute the second derivative (hessian matrix) of the mahalanobis distance $$ [x-\mu]^{T} \Sigma^{-1} [x-\mu] $$ wrt to the Cholesky decomposition of ...
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10 views

Chemical Reaction Networks: Applications

I have recently read a bit on chemical reaction network theory. I was wondering whether the mathematical concepts have cross field applications like neural networks. For example, can I apply chemical ...
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9 views

How to linearize the multiplication of optimization variables?

I have to linearize the objective function of my optimization problem which is in the form: $$\text{Maximize }\prod_{n=1}^{N}x_n $$ where, $x_n\ge 0$ and $N>2$.
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27 views

Sum/product of two functions of two variables are to be minimized

I have two functions $f(x,y)$ and $g(x,y)$ whose sum/product (whichever is possible) is to be minimized. The values of $x,y$ can vary in the interval $0<x,y<1$ (hence none of them can have a ...
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1answer
31 views

Least Squares Sensitivity to data

Let ($x_1$,$y_1$),...,($x_n$,$y_n$) be my data set. I have a function $f(x,{\bf c})$ where ${\bf c}=(c_1,...,c_m)$ is a vector of $m$ parameters. I want to fit to the data using non-linear least ...
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A convex optimisation problem involving the Euclidean norm

Any ideas on how to approach the following optimisation problem? $$\begin{array}{ll} \text{maximize} & \|Ax\|_2^2+\|Bx\|_2^2+\|Cx\|_2^2 \\ \text{subject to} & \|x\|_2 = 1\end{array}$$
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Quasiconvex objective function with nonlinear constraint

I have the following optimization problem: $$\min \frac{F}{A}+c_p D K \sqrt{A}+c_g \Theta^2+c_e a D + t e (1-\Theta) D+ \frac{t a}{A}$$ s.t. $a \leq [e (1-\Theta) D A] p_a$ $$\Theta \leq p_\Theta$$ ...
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30 views

Lasso with non-linear objective

I have a non-linear objective function that I want to minimize considering some constraints in order to obtain a sparse solution (lasso type). min f($\theta$) s.t. $\sum_i|\theta_i|\leq t$ $\theta_i ...
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1answer
22 views

floor/ceiling/round functions in the constraints of an optimization?

I have a constrained optimization problem in which I have to impose a "floor" or "ceiling" constraint to the solution. In fact I decided to use these nonlinear rounding functions because I needed to ...
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2answers
34 views

Change of variables in minimization

I have the following non linear programming to solve: $$\left\{\begin{matrix} \min & (x-y)^2 +e^z+e^{-z} \\ \text{s.t.} & xz=0 \\ & yz=0 \end{matrix}\right.$$ The book suggests to ...
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54 views

How can I experiment with Lagrange multiplier in QCQP?

Suppose we want to solve following optimization problem (it is a PCA problem in this post) $$ \underset{\mathbf w}{\text{maximize}}~~ \mathbf w^\top \mathbf{Cw} \\ \text{s.t.}~~~~~~ \mathbf w^\top \...
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45 views

Why the original MINLP and Linearized MILP are giving mismatched results?

I have an MINLP and its linearized formulation problem given below where the objective (nonconvex) and constraint C4 are nonlinear. We linearized them by applying some known techniques. However, when ...
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24 views

Book Recommendation for Infinite Dimensional Stochastic Optimization Problem in Discrete Time

Let $X(k)$ be i.i.d. discrete random variables and for all $k=0,1,2,...,N-1$, let $X:=(X(0),X(1)...,X(k))$ and $f := (f(0), f(1),...,f(k))$ with $f(k)$ be the decision function at time $k$, I want to ...
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48 views

Minimizing a strictly convex function with inequality constraint

So we've been learning about the Kuhn Tucker conditions in my non-linear optimization course and I've been having trouble with this problem: QUestion: description here Question: a strictly convex ...
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157 views

Optimal decomposition of discrete function into sum of factorised terms

I am trying to solve the following optimisation problem. Let $x_i \in \{1, \ldots, N_i\}$ be discrete variables, and $f(x_1, \ldots , x_n)$ any real-valued function. I want to decompose $f$ into a ...
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24 views

What does it mean if the KKT conditions do not result in a real solution for a convex problem?

Given a convex optimization problem with equality and inequality constraints, the KKT conditions are sufficient and necessary conditions for optimality. What does it mean if the KKT conditions do not ...
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21 views

modelling a composite objective function (max + argmax) as an (integer) linear program

Suppose $\mathbf{x} = [x_1, x_2, \ldots, x_n]$, where $x_i \in \{0, 1\}$ are binary variables. We know for a fixed $\mathbf{w}$ the following problem is an Integer Linear Program: $$ \arg\max_{\...
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31 views

Smallest enclosing cylinder

I have a set of 3D points that approximately lie on a cylinder. This cylinder is straight and can be oriented in any direction. I would like to compute the minimal enclosing cylinder for the set; that ...
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21 views

linearize average success probability constraint

In my optimization problem, I've a constraint to calculate the average success probability of a path. $x_{i,j}$ is binary variable defined as: $$ \begin{align} \label{eq3:1} x_{i,j} = \begin{cases} ...
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183 views

Optimize nonlinear model over time period (w/ Kuhn-Tucker Conditions)

I'm working on a nonlinear model that includes a time interval of 12 months. The goal is to maximize the total net benefit (NB) over the entire time period given the constraints listed below. I've ...
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35 views

Maximum of Contour Line

Consider the potential $U(x,y)=ay^{2}+b(e^{x-y}-1)^{2}+c(e^{x+y}-1)^{2}$ where $a$, $b$ and $c$ are known constants. I want to move through a contour line of this potential $U(x,y)=k$, say $y=g(x)$. ...
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21 views

Non linear functional optimization under constraints

For some given positive functions $l(t)>0$ and $h(t)>0$, such that $h(t)>l(t)$, I want to solve this functional optimization problem on $a(t)$: $\min_a\int_0^T[l(t)\cdot\min(a(t),0) + h(t)\...
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33 views

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution?

Does $\min\{x_1x_3+x_2^2 | x_1^2+x_2^2+x_3^4 = 4\}$ has an optimal solution? I think continuous function over closed and bounded domain has an optimal solution but I am not sure. Can anyone give me ...
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1answer
26 views

Solving an optimization problem with a linear objective and quadratic constraint

The title is general, but what I am specifically interested in, is how to solve the following problem: $$\text{Maximize } c $$ $$\text{Subject to:}$$ $$a+b+c<0$$ $$b^2-4ac<0$$ $$a,b \in \...