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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1answer
756 views

Python numerical solution for a nonlinear second order ODE with two boundary conditions

I want to solve numerical the next equation, in Python $$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$ it is a nonlinear second order $ODE$ with two $B.C$. ...
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2answers
55 views

How do I derivate a function in the Norm?

I have a funtion $$f(x) = \frac12\|G(x)\|_2^2$$ where $G(x): \mathbb R^n \rightarrow \mathbb R^n$ is a twice continuous differentiable funtion. I want to determine $\nabla f(x)$ in terms of $G(x)$ ...
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2answers
81 views

Anyone saw this interesting function before?

Say $\theta\in\Re^n$ and $\theta_i\in(0,1)$ for all $i$. Define $$ f(\theta) = \frac{1}{n}\sum_i^n\{(1-\theta_i)\log(1-\theta_i)+\theta_i\log\theta_i\} $$ It is easy to see the minimizer of ...
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1answer
911 views

Rate of Convergence for Gradient Descent (Example)

I am trying to determine the rate of convergence for $f(x,y) = 5x^2 + 5y^2 − xy − 11x + 11y$. Would anyone be able to provide guidance as to how I might go about doing this? Should I select my own ...
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1answer
215 views

Nonlinear method to solve an equation with the error function in it

My question is to find a method to solve the following non-linear equation. I know it should be an iterative method, but I don't know what would be the best method to use. Any help is highly ...
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0answers
62 views

Minimum of some functions

Denote $U=\{(x_1,x_2,...,x_n):0<x_j<1 (1\leq j\leq n),\sum_{j=1}^nx_j=1\}$. Let $f_i=f_i(x_1,x_2,...,x_n)$ ($1\leq i\leq n-1$) be $n-1$ real functions which satisfy: ...
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1answer
60 views

Distributed Robust Optimization

Consider the following constrained optimization problem $\mathcal{P}$. $$ \min_{x \in X \subseteq \mathbb{R}^n} f(x) \ \text{sub. to: } g(x,y) \leq 0 \ \forall y \in Y \subseteq \mathbb{R}^m $$ ...
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1answer
100 views

Distance between a point to a $2d$ ellipse in $3d$ ambient space

Suppose we are working in the 3D Euclidean space. We are given an arbitrary point $p$ and a 2d ellipse: $$E=\{x:x^TQx\leq1,x^Tq=0\},$$ where $Q$ is a positive definite matrix and $q$ is an ...
2
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0answers
54 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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1answer
339 views

Lipschitz constant for optimization of multivariate function

I intend to implement an optimization algorithm which requires the computation of the Lipschitz constant. My function is a multivariate function with more than 50 variables. I am wondering whether ...
3
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1answer
681 views

Lagrange Multipliers for Function Spaces

For some constant $A > 1$ I am trying to solve the constrained minimization problem minimize $F(u)$ in $C$ subject to $H(u) = 0$. Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
3
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1answer
33 views

existence of solution of $Ax= \max(b-x,0) $

How do you prove the existence of a solution to the linear system: \begin{equation} Ax= \max(b-x,0) \end{equation} A is an $n\times n$ matrix and $b$ is a vector in $\mathbb{R}^n$. $x$ is the ...
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2answers
157 views

optimality of quadratic programming problems

Suppose we have a general quadratic programming problem: \begin{align} \min_{x}\,\,&c^Tx+\frac{1}{2}x^TQx,\\ \mbox{s.t.}\,\,& Ax=b,\\ &x\geq0, \end{align} where $Q$ is positive ...
2
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1answer
279 views

Sion's minimax theorem

Sion's minimax theorem is stated as: Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. Let $f$ be a real-valued function on ...
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0answers
41 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 $c \mu - \log\det(B)$ ...
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3answers
669 views

Finding good approximation for $x^{1/2.4}$

I would like to a good (8 bits accuracy) approximation for $x^{1/2.4}$ in the range $[0, 1]$. This transform is used for converting linear intensities to SRGB compressed values, so it's important that ...
2
votes
1answer
425 views

Numerical/artifical damping in forward Euler?

I'm testing a code to find periodic solutions of nonlinear structural vibrating systems by solving a global time-discretized periodic system of equations. I am using a forward Euler (first order ...
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2answers
376 views

Maximize the product of linear functions

Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$ where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum. What's the most efficient ...
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1answer
84 views

Nonlinear Systems- L2 stability analysis

I hope you are having a good day. I am working on a homework and I was looking for some help. Can anyone please help me with the next step to prove whether the L-2 stability of the system and the ...
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1answer
205 views

Check Kuhn-Tucker conditions

How to check if $(0,1)$ point is the solution of this optimization problem using Kuhn-Tucker Theorem. Find the min of $e^{x_1-x_2}-x_1-x_2$ where $x_1+x_2\le1,\ x_1\ge 0,\ x_2\ge0$ I am thinking ...
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1answer
63 views

Find a decoupled explicit formula for a minimizer

Consider the energy $F(u,v) = \int^1_0((\frac{1}{4}(u')^2+(v')^2 +\frac{1}{2}(u-v+1)^2)dx$ for $C^1$ functions u and v on the interval (0,1) that satisfy the boundary conditions ...
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2answers
182 views

Finding an explicit expression for a minimizer

Suppose $f$ is a continuous function on the interval (0,1). We consider the energy functional $F(u) = \int^1_0\frac{1}{2}((u')^2+u^2)\,dx - \int^1_0fu\,dx$ which is well defined for continuously ...
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0answers
279 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 ...
0
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1answer
57 views

non linear equations solving methods?

I need to find $l_{2}$ and $\theta$ numerically by solving below equations. How could I do that? At least do i have some iterative way of finding those two unknowns. All others parameters are ...
0
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1answer
396 views

Convex Functions: Property Proof

Let $f\colon S\to \mathbb R$ be a $C^1$ function on a convex domain $S \subseteq \mathbb R^n$. Show that if $f$ is convex then $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \ge 0$ for all $x,y \in S$. ...
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1answer
61 views

Smooth Reformulation of NonSmooth Constraints

If I have something like : \begin{align} \min_x \max_i f_i(x) \end{align} I can reformulate this nonsmooth formulation as: $$\min_x z$$ $$z\geq f_i(x)$$ and I have a smooth formulation of the problem. ...
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0answers
370 views

Matlab quadprog lower bound constraint ignored?

I am trying to solve an optimization problem with matlabs quadprog function. I have set up the problem to solve only equality constraints with lower bounds of 0. ...
3
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2answers
164 views

Upper bound for Maximization problem

I have an optimization problem of the form Max $x_1+x_2+x_3+\cdots+x_n$ subject to $x_0^2+x_1^2+x_2^2+\cdots+x_n^2+x_{12}^2+x_{13}^2+x_{14}^2+ \cdots+x_{1n}^2+x_{23}^2 + \cdots +x_{2n}^2+ \cdots ...
1
vote
1answer
241 views

Convex Functions: Proofs

Let $f$ be a monotone nondecreasing function of a single variable which is also convex. Let $g$ be a convex function defined on a convex set $G$. Is it true that the composition of these functions ...
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0answers
78 views

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 ...
1
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1answer
887 views

is nonlinear least square a non convex optimization?

linear least-squares are convex optimization. Are nonlinear least squares also convex optimization? Can someone please give some simple examples?
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0answers
56 views

Implementing Non Linear Optimization

I am trying to calculate model free implied volatility $\sigma_{\mathrm{MF}}$ for a relative performance index using the following method: $$ \sigma_{\mathrm{MF}}^2=2\sum_{i} ...
2
votes
1answer
177 views

non linear optimization

How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in ...
2
votes
1answer
139 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
3
votes
1answer
237 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
1
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1answer
559 views

Constrained Optimization - Lagrange Multipliers (Example)

Let $f(x, y, z) = xyz$ $h1(x, y, z) = x + y + z − 4$5 and $h2(x, y, z) = 2x − y$. Goal: Minimize $f(x, y, z)$ subject to $h1(x, y, z) = 0$ and $h2(x, y, z) = 0$. First part: Show that every ...
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1answer
738 views

Prove every local minimum is a global minimum

Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$. Minimize $$f(x)= ...
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1answer
2k views

The composition of two convex functions is convex

Let $f$ be a convex function on a convex domain $\Omega$ and $g$ a convex non-decreasing function on $\mathbb{R}$. prove that the composition of $g(f)$ is convex on $\Omega$. Under what conditions is ...
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0answers
171 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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1answer
2k views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
3
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1answer
86 views

$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)

This problem arose in my stereo vision project. $$ P_{1c} = A*P $$ $$ P_{2c} = B*P $$ where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
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4answers
540 views

Approximate a function over the interval $[0, 1]$ by a polynomial of degree $n$ (or less).

To approximate a function $G$ over the interval $[0,1]$ by a polynomial $P$ of degree $n$ (or less), we minimize the function $f:R^{n+1} \to R$ given by $F(a) = \int_0^1 (G(x) - P_a(x))^2\,dx$, where ...
2
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1answer
117 views

Nonlinear optimization question

For (x,y) in $\mathbb R^2$, consider f(x,y) = $x^2 -2xy + \frac{4}{3}y^2 - 4y$ Find the local minimum of f. Is it a strict local minimum? Compute the $\lim\limits_{|(x,y)|\to \infty}$ f(x,y) to decide ...
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60 views

Deconvoluting linear combinations of unknown distributions

Summary I am trying to deconvolute the distribution $T(x)$ of a population's $x$ parameter into sub-distributions ($P(x)$, $Q(x)$, $R(x)$ ...), of which I don't know the form (only that they have ...
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1answer
65 views

Reformulation of BQP to SDP

I run into the following reading some optimization papars: $$\min_x x^TAx $$ where $x\in\{-1,1\}^n$ and $A\in S_n$, Is equivalent to $$ \min <X,A>$$ s.t $diag(X) = (1,1,...,1)\;\; rank(X) = 1$. ...
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0answers
18 views

Existence of solutions to a Specific kind of non-linear system of equations with rational variables

Consider a system of the form $c_{11}a_1b_{1} + c_{12}a_2b_{2} + \cdots c_{1n}a_nb_{n} = q_1$ $c_{21}a_1b_{1} + c_{22}a_2b_{2} + \cdots c_{2n}a_nb_{n} = q_2$ $\vdots$ $c_{m1}a_1b_{1} + ...
4
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1answer
186 views

Kuhn-Tucker condition is not satisfied

Show that the solution to finding minimum of $f(x)=-x_{1}$ With conditions $-\sin(x_{1})+x_{2} \leq 0$ $x_{1}-x_{2} \leq 0$ is point $(0,0)$, but the Kuhn-Tucker condition is not satisfied in this ...
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0answers
63 views

why is it important to have $\max_x \min_y f(x,y)=\min_y \max_x f(x,y)$?

I am currently trying to understand the minimax theorem of Von Neumann and the improved versions of this theorem. At any case we have the property $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} ...
0
votes
1answer
128 views

Minimization to Maximization doubt in SVM

I came across a lecture on Support Vector Machines and in the lecture they converted a maximization problem into a minimization problem. I am wondering how it was done... $ Max \frac {1}{||x||} $ ...
0
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2answers
151 views

Minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$

How can we minimize $z=2x_1+3x_2-x_1^2-2x_2^2$ subject to $x_1+3x_2\le 6$, $5x_1+2x_2\le 10$, and $x_1,x_2\ge 0$? I need to know the steps to solve or at least the guidelines as I am really new to ...