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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.

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Pontryagin Maximum Principle - Mayer form - Null adjoint

Consider an optimal control problem written in a Mayer form $$ \min_{u(\cdot)\in L^\infty([0,T])} \varphi(x_u(T)) $$ where the state $x_u\in \mathbb{R}^n$ associated to the control $u$ satisfies $$ \...
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Handling singular matrices in gradient-descent optimization.

Right now I am coding up optimization for a 70 dimension nonlinear optimization, where the analytical gradient is unavailable. I have some non-linear constraints that maps the structural parameters ...
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What is the smallest time complexity to solve this optimization problem?

I want to find the best way to solve this optimization problem in the smallest time complexity: Given $$n, C, r_i, p_i,a_i \quad∀ i={1,2,...,n}, $$ $$maximize \quad f(x_1,x_2,...,x_n)=\prod_{i=1}^n {...
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Optimization problem with constraint containing one design variable and one non-design variable

I have an objective function $f(x_1,x_2)$ which should be minimized while a function $g(x,x_1,x_2)$ MUST be greater than 0.8 for ALL x in an intervall $x=[0,10]$. As you can see, the constraint has a ...
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Minimizing sum of reciprocal of quadratic functions

Given a set of constants $a_1,\ldots,a_n$, I want to solve the following single-variable optimization problem: $$\min_x \sum_{i=1}^n \frac{a_i^2}{x(2a_i-x)}, \quad s.t. \quad 0\leq x \leq 2a_i, \...
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*Recovering* Lagrange multipliers

I am currently reading this paper : https://www.di.ens.fr/~fbach/skm_icml.pdf about Multi-instance kernels, and in section 2.1 (page 2) it is written the coefficients $\eta_j$ are recovered as ...
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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?
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Dual of an SDP ${\min}_{X \in \mathcal{S}^n} \quad \ {\rm trace}( W X )$ s.t. $X_{ii} = 1$; $X \succeq 0$

How to obtain the dual of the following semidefinite programming problem (SDP) \begin{align} \text{minimize}_{X \in \mathcal{S}^n} \quad & {\rm trace}( W X ) \\ \text{subject to }\quad & X_{...
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Differences between objective function definitions in optimization problems

Can someone explain the differences between the relative error definitions used as an objective function in a constrained optimization problem? I am trying to understand why I get different values of $...
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Find minimizer of mean values

i'd like to know if there is an analytical method to solve the following optimisation problem : $\forall i=1,..,n$ find $\omega_i^{}$ and $\alpha_i^{}$ such that: $\dfrac{1}{n} \displaystyle \sum_{i=...
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How can I mathematically prove this?

How can I mathematically prove that P1 and P2 will have the same value of $\eta$ at optimality? Although it seems clear from the intuition. I am looking for proof in the language of mathematics, not ...
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Is there efficient Surface Walking method for optimization problems with equality constraint?

To my best knowledge, if we want to find the minimum of a function $f$ defined on a $d$-dimension manifold $M$ in $\mathbb{R}^n$, a.k.a an optimization problem with equality constraint, the most ...
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Optimize placement to yield Shortest Distance to 20 Points

Hey guys looking for a maths wiz or IT expert to help me solve this problem. I have 20 points (say 20 people) in a 2D plane (X,Y) all the coords are given for each of the 20 people. I would like to ...
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Changing from non linear to linear using Big M

I have a binary variable $v_{ij}$ and integer variable $c_{ijk}$ and the following relationship: $$ c_{ijk} \le M \; v_{ij} $$ $M$ is a very large number Is there a way to change this nonlinear ...
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The Dual Model of Primal Non Linear

I am working on the follwoing nonlinear model. Min z=10(1-$\exp(-3x)$) subject to: x $\leq $ 3 How can I build the dual model of this nonlinear model ? Thank you in advance
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Convex Optimization: objective function in minimum cost flow problem

I have a graph $G=(N,V)$ where $|N|=n$ and $|V|=m$. I want to implement a solver (based on a specific algorithm for convex optimization) for a convex quadratic separable Min-Cost Flow Problem. $$min\{...
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How to linearly approximate the L-2 norm of a complex vector?

The complex vector is given by ${\bf x}=[x_1,x_2,\cdots,x_k]$. How to linearly approximate the $l_2$-norm of $\bf x$?
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Prove that $\xi_{k+1} = (-1)^{k+1}(\alpha_0 \times \alpha_1 \times \dots \times \alpha_k)$ is the (k+1) coefficient of $p_k$

I was given the following question as part of a homework assignment. Any help would be greatly appreciated! The following image shows the steps of a preliminary version of the conjugate gradient ...
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How to linearize this objective funtion?

The objective function I am dealing is $$\underset{{\bf w}_k,x_k }{\max}\sum_{k=1}^K x_k\alpha_k \log_2(1+\gamma_k)$$ subject to $\sum x_k ||{\bf w}_k||_2^2\le P$ and $\sum_{k=1}^Kx_k=L.$...
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How to linearize this IF-THEN Constraint?

I have a nonlinear constraint as below If $x_k=0$, Then $||{\bf w}_k||==0$ If $x_k=1$, Then $||{\bf w}_k||>0$ Here, $x_k\in\{0,1\}$ is a binary variable and $||\bf x||$ is the norm of vector $\...
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How to linearize these constraints in scheduling optimization problem?

I have a mixed integer programming problem as below $$\underset{{\bf w}_k }{\max}\sum_{k=1}^K x_k\alpha_k \log_2(1+\gamma_k)$$ subject to $$\sum_{k=1}^K x_k||{\bf w}_k||^2_2\le P$$ $$x_k\in\{0,1\}$$ ...
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Gradient optimization with attractors/preferences, that respects already emerged structures in parameter subspace?

One can assume that certain structures have emerged in the space of parameters which are being optimized to achieve some minimum of some function over those parameters, e.g. f(p1, p2, ...). By ...
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Duality gap in non-convex optimization: Do KKT conditions+constraint qualification imply strong duality?

Consider the non-convex optimization problem: $$\underset{x\in \mathbb{R}^n}{\min} ~f(x) \\ \mbox{s.t.}~h_i(x)=0 ~~~\mbox{for}~~~i=1,\ldots,p \\ ~~~~~~ g_j(x)\leq0 ~~~\mbox{for}~~~j=1,\...
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Maximise $p_1p_2p_3p_4p_5$ subject to constraints

Given $x_5 \geq x_4 \geq x_3 \geq x_2 \geq x_1 \geq 0$, solve the following optimization problem in $p_1, p_2,\dots, p_5$. $$\max p_1p_2p_3p_4p_5$$ subject to: $$p_1 x_1 + p_2 (x_2 - ...
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Lagrange multipliers with inequality constraints

When solving non-linear optimization with inequality constraints, one method seems to be to divide the problem in two parts and solve it inside the boundary and on the boundary. My question is: why do ...
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Constrained Optimization for Location Allocation problem- new supply center selection for maximal coverage of demand points

I am trying to figure out the objective function and associated constraints for optimization of the following problem: There is a set of Demand points I, which have to be covered by a set of Supply ...
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How to linearize the product of a non-binary discrete variable and a continuous variable?

Given a set $J$, I have the following constraint: $w_j = y_j u \quad \forall j \in J$ where $y_j \in \mathbb{N}$ and $u \in \mathbb{R}⁺_0$. I would like to make this constraint linear. Note: I am ...
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Minimization of piecewise real multivariate function

I want to minimize a multivariable function with the following form: $$ f(\vec{x}) = g(\vec{x}) + | h (\vec{x}) | , $$ this is a piecewise function depending on the sign of $h (\vec{x})$, and I assume ...
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1answer
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Complexity of first, second and zero order optimization

I am currently reading Bishop - 'Pattern Recognition and Machine Learning' (2006) where he writes about why using gradient information for optimization is superior to not using it. (p. 239) ...
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Convex approximation of the non-convex function

I have the following constraint for the optimization problem in hand: $$\frac{b_k}{\log_2 \left(1 + \frac{p_k \alpha_k}{\sum_{j \neq k} p_j \alpha_j + \sum_n \sum_l \Xi_{n,l,k} 2^{-2 v_{n,l}}} \...
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1answer
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Age matching optimization problem.

A common problem in clinical studies is how to choose age-, and usually also gender-, matched pairs from two groups such as case/control. (case=has disease, control=not have). After searching the ...
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Optimize a fractional multivariate function

Consider the following optimization problem: $\max\limits_\mathbf{w} \frac{\frac{\sum_i w_i}{2} - \sum_i w_ip_{i}}{(\sum_i w_i(1 - p_{i})p_{i})^\frac{1}{2}}$ s.t. $\mathbf{w} = [w_1, w_2,..., w_M ]^T ...
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Gradient of $L(W_1, W_2, W_3) := \sum_{i=1}^N \| W_3 \ g\left(W_2 \ f\left(W_1 x_i \right) \right) - y_i \|_2^2 + \lambda ( \sum_l \| W_l\|_1)$?

Extending this question. How to obtain the gradient of ($\ell1$ penalized) \begin{align} L(W_1, W_2, W_3) := \sum_{i=1}^N \| W_3 \ g\left(W_2 \ f\left(W_1 x_i \right) \right) - y_i \|_2^2 + \lambda \...
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Lagrange Multiplier quadratic with positivity constraint

Is it possible to solve the following problem using Lagrange Multipliers? If not Lagrange multipliers, what is the best way to approach this? Maximise $(p - \frac{1}{4})^2 + (q - \frac{1}{4})^2 + (r -...
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MINLP problem with linear constraints and piecewise term

I have an optimization problem where the objective function is non-linear but (I think) differentiable, there are linear constraints, and all parameters are integers. To make matters more difficult, ...
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Maximizing $x^T A^T B \, x$ over the unit Euclidean sphere

Is there an algorithm to solve the QCQP $$\begin{array}{ll} \text{maximize} & x^T A^T B \, x\\ \text{subject to} & \|x\|_2 = 1\end{array}$$ when $A^T B$ is not necessarily symmetric? When $A^...
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How to reduce number of nonlinear terms in nonlinear equation by linear substitution?

Consider system of nonlinear equations: $2a-2b-3=0$ $a-a^2-b-2ab-b^2=0$ If you replace variables this way: $x=a+b$, $y=a-b$, then this system can be simplified to $2y=3,y=x^2$. My question is ...
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How to do a sensitivity analysis on a non-linear equation?

In the company, it is very difficult to actually do quotations for our customers properly because we do not have perfect information regarding the factors that affect the cost and profit. So I created ...
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How to set grad[ ] in nlopt?

I would like to ask you about NLopt as follows below. Question1: If the number of constraints is bigger than the number of variables, how can we set “grad[ ]” in the “myconstraint”? Is there any (...
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Error estimate in the approximation of Incomplete Beta Function

In studying a problem related with the negative binomial, I need to approximate some function, which is a polynomial combination of the incomplete Beta function, in this case given by $$f_{a,b}(x):=1-...
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No feasible direction increases the objective function, yet this point isn't a local maximum?

Consider the optimization problem of maximizing $x$ subject to $x \leq 5$ and $\sqrt{x^2 + y^2} \geq 1$. if you examine the graph of the feasible region, you will see that all feasible directions for ...
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Can I still do the normalization, or how do I do it here?

I have to following optimization problem, $$\max_{T} \left[ e^{\eta _{0}}\left( W_{h}+1\left( \eta _{t}>0\right) b-T\right) ^{r}+\psi T^{r}\right] ^{1/r}+\eta _{h}1\left( \eta _{h}>0\right) $$ ...
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Finding a global minimum

I seek the function $f$ which satisfies the 100 equations (i=1,2...100) $\sum_{j=1}^{2000} f(A_{ij},B_{ij},C_{ij})=Q_i$. Where $A,B,C$ are 100x2000 matrices and all entries are between 0 and 1. ...
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Non-Linear to Linear with Auxiliary Binary Variables

The Problem: Formulate the constraint $u = \min\,\{x_1, x_2\}$ with linear constraints and binary variables. We assume that $0 \leq x_i \leq 10$ for $i = 1, 2$. Specify the value of every big-M ...
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About the maximum distance between a point on a trajectory of a dynamical system, and its projection onto its linear interpolation

Summary This is a question regarding the maximum error that sampling of a dynamical system trajectory introduces w.r.t. the chosen time-step $\delta$. I formulate this as an optimization problem of ...
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Nonlinear Optimization: Explain how to differentiate a norm of a nonlinear function using matrix algebra

I am learning about nonlinear optimization where I have some data vector, $\mathbf{d}$ and some unknown model $\mathbf{m}$. I can calculate predicted data from some model using a nonlinear function $\...
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How to use duality in optimization?

I understand the concept of duality in convex optimization. However, I am not able to understand how we can use it to solve problems. Primal problems can be directly solved using Newton's method or ...
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Unconstrained Optimization Introduction Question on Youtube

I'm looking at this youtube introduction on unconstrained optimization for one variable at this point in time and it says that if: f(x1) < f(x2) then we narrow the search range to [a, x2] instead ...
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Total Unimodularity in Integer QP

I have a Quadratic Programming problem with linear constraints. My objective is Quadratic-Convex, the constraint matrix is Totally Unimodular (similar to assignment or network flow problems), and the ...
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Minimal eigenvalue of hessian matrices

I've got a problem that seems to be wrong to me. Some clarification would be of great help! Let $f:\mathbb{R}^n \to \mathbb{R}$ be twice continuously differentiable with local minimum $\overline{x}$ ...