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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Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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202 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
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28 views

Nonlinear optimisation of Expectation

I am preparing for my exams and I can't get my head around the following question. I know there exists a general method for solving these problems but I don't know where to start. I would greatly ...
4
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152 views

Why Compactness is Necessary at Minimax Theorem

According to Von Neumann's minimax theorem, I have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
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295 views

(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
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135 views

Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
3
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35 views

Solve non-linear equation with Matrices

I'm looking more for hints than specific answers, although I would be extremely grateful if provided with one. The problem I have is as follows: $$ -\Sigma (A+\Lambda_1)+I=0 $$ Here A is a constant, ...
3
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45 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
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24 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
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72 views

Solver for sparse linearly-constrained non-linear least-squares

Reposted from stackoverflow on the advice of Nick Rosencrantz: Are there any algorithms or solvers for solving non-linear least-squares problems where the jacobian is known to always be sparse, and ...
3
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269 views

Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
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17 views

discrete nonlinear convex optimization relaxation over a dense set

Be a discrete nonlinear convex optimization problem $P$ \begin{align} \underset{x\in \mathrm{C}^n}{\mathrm{min}} \ \ \ f(x) \\ Ax=b \\ c \leq x \leq d \end{align} $C$ is a dense in $F$. Is solving ...
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33 views

Exercice the converge $r$-superlinearly

Give $x_0 \in \mathbb{R}^*$. Show that $\{x_k\} \subset \mathbb{R}$ converge $r$-superlinearly for $x^∗=0$, where $x_k$ is defined by $x_{k+1}=(1−\beta_k)x_k$ and $\beta_k=1−2^{-k}$ if $k=i^2$ for ...
2
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48 views

Constrained non-linear Optimization using Newton's method - Portfolio optimization

I want to solve following constrained optimization problem from portfolio optimization: The solution is supposed to be a modified risk parity portfolio: The optimization problem is: \begin{align} ...
2
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37 views

Constrained Quadratic Optimization(Reproducing Kernel)

I am attempting to use a constrained quadratic optimization to find the coefficients of a reproducing kernel. The problem is as follows: $y(t)=\sum_{i=0}^J\alpha_iK(t, t_i)$ $Q(\alpha)= ...
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100 views

Levenberg's original article “A method for the solution of certain problems in least squares”

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168? It is ...
2
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78 views

Scaling factor and weights in Unscented Transform (UKF)

I'm trying to implement the UKF for parameter estimation as described by Eric A. Wan and Rudolph van der Merwe in Chapter 7 of the Kalman Filtering and Neural Networks book: Free PDF I am confused by ...
2
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97 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
2
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37 views

Calculating second derivative of $g(\alpha) = f(\textbf{y}(\alpha))$

I'm having problems with the second derivative of the function $g(\alpha) = f(\textbf{y}(\alpha))$ (which I will define more precisely below). I tried calculating it myself, could anyone just simply ...
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59 views

the objective function $\|F\|_F^2$ is quasiconvex in the optimization?why?

I have read a paper, but I can not understand one optimization thoroughly.Generally, Frobenius norm of one matrix, $\|F\|_F^2$, as the objective function is convex, so we can resolve it not using the ...
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73 views

Steepest Descent/Newton

Suppose these over-determined system of equations: $$ |\mathbf{x}^T\mathbf{v_n}| = A, \qquad n = 1,2,\cdots,N-1 $$ $$ \mathbf{v_n}= [1 \quad w^n \quad w^{2n} \quad \cdots \quad w^{(N-1)n}]^T , ...
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72 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
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70 views

Nonlinear optimization of constraint parameter - subdifferential?

Disclaimer: I discovered that the FAQ suggests to post research-level to mathoverflow instead of math.stackexchange. I "moved" the question accordingly, cp. post at mathoverflow. Sorry for the ...
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339 views

SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...
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53 views

Linearization of a dynamic system, $\small 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $

I have a dynamic system, with in signal u and out signal y. The system is described with the following differential equation $$ 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $$ The task is to ...
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38 views

Convex formulation of a nearly convex optimization problem

The following problem has come up in my studies of logarithmic norms. I wish to find $\mu \in \mathbb{R}$ and a positive semidefinite $B$ so as to minimize the convex function $c \mu - \log\det(B)$ ...
2
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267 views

sufficient condition for KKT problems

For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that: "The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
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216 views

Optimization over function spaces

There are many methods to solve optimization over standard spaces such as real and complex numbers or vector spaces of these. In my case I have a vector space of functions of a real value to find. ...
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670 views

Convex minimization over the Unit Simplex

I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq ...
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15 views

Optimal value of decision variable leads to inconsistency

$\epsilon$ is a random variable with support in $(0.8,0.95)$ and pdf $f(\epsilon)$. The following equation arises out of a business problem: $ENP=800*A*E(\epsilon)+ 9000 - ...
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25 views

Minimization using logarithmic barrier function

I'm thinking of the quadratic problem(QP) \begin{align} &\underset{x\in \mathrm{R}^n}{\mathrm{Minimize}}\ \ \ \frac{1}{2}x^\top{}Qx + f^\top{}x\\ &\mathrm{subject\ to}\ \ \ \ a_ix \leq b_i\ ...
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23 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
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Finding a Sparse Binary Vector Which Satisfies a Vector Ordering

I have a research problem that I have reduced to the following problem which seems to be hard: Find $\mathbf{x}\in \mathbb{B}^N$ $(\mbox{where }\mathbb{B}= \{0,1\})$ which is $K$-sparse and which ...
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27 views

Newton's method: Is the change of parameter values between consecutive steps always decreasing?

Assume that I have a twice differentiable function $f(x)$ which I try to maximize with respect to $x$ (let's say $x$ is $k$-dimensional vector). When performing optimization via Newton algorithm, ...
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26 views

Are there any algorithms for simultaneous optimization of multiple objective functions?

I would like to minimize a set of similar objective functions $$f_\boldsymbol{s}(\boldsymbol{x}),$$ where $\boldsymbol{x} \in A \subseteq \mathbb{R}^M$ and the parameterization $\boldsymbol{s} \in S ...
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22 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
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49 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
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38 views

Line search Armijo, Wolfe, Strong Wolfe and Goldstein.

What are the articles (References) who proposed the line search of Armijo, Wolfe, Strong Wolfe, and Goldstein? Articles precursors of unidirectional searches?
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48 views

Proving/deciding concavity of a function of two variables

I would like to formally prove that the function $f(x,y) = \frac{(c+1)e^{-x}(xe^{x+y}+y)}{(c+2)(e^{x+y}-1)+e^y} $ is concave ($ c>2$ is a constant, and both $x,\, y \in \mathbf{R_+}$). Plots of ...
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30 views

When result of max of min problem is equal to min of max problem

Let's assume there are two functions $f(x)$ and $g(x)$. I want to know when the optimal $x$ of max of min of $f(x)$ and $g(x)$ is not equal to optimal $x$ of min of max of $\frac{1}{f(x)}$ and ...
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25 views

Explain KKT conditions without reference to duality.

Is it possible to explain (not derive) KKT necessary conditions without reference to the concept of Lagrangian duality?
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25 views

Solving many independent non-linear systems simultaneously

I'm working on solving lots of systems of nonlinear equations. Luckily, the non-linear equation is the same, but the parameters are different: $$ f(\vec{x}_0; c_0) = 0\\ f(\vec{x}_1; c_1) = 0\\ ...
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How can I find the unit vector that minimizes the number of nonzero projections that a set of points has on it?

$\underset{\mathbf{w}}{\min} ~ \|\mathbf{X}^T\mathbf{w}\|_1~~~\text{subject to:}~ \|\mathbf{w}\|_2^2=1$ where $\mathbf{X}\in\mathbb{R}^{d\times m}$ is a set of $d$-dimensional points and $m>d$. ...
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25 views

Non-convex constraint made cost

Consider the non-convex optimization problem $$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$ where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are ...
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27 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
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34 views

Gradient descent via polynomial approximation

It seems that most proofs of convergence for gradient descent algorithms rely on strong conditions on the first and second derivatives of the function, for instance that $$|f''(x)| \leq K$$ over the ...
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27 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
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30 views

Zero-order necessary conditions

I have a question regarding the Zero-order necessary conditions. In my Linear and Nonlinear programming book it is stated: Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): r\geq ...
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61 views

implicit non-linear equations with complex variables

I am trying to understand a methodology for solving implicit non-linear equations with complex variables. I would like to solve for z1 below where z2 is known. Also both z1 and z2 are complex ...
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50 views

Strong duality in trace maximization

I'm working on understanding the derivation of the solution for principal components analysis. Let $\mathbf{S} \in \mathbb{R}^{p \times p}$ be a positive semi-definite matrix with rank $d < p$. ...