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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Scaled proximal operator for proximal Newton method

The scaled proximal operator was introduced as an extension of the (regular) proximal operator: $prox^H_h(x) = \arg\min_y h(y) + \frac{1}{2}\|y-x\|^2_H$. (See ...
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25 views

How to solve the following convex constrained optimization problem?

\begin{equation}\label{constrained optimization} \begin{aligned} \min\limits_{\mathbf{X}}&\|X_{(1)}\|_{*}+\|X_{(2)}\|_{*}+\|X_{(3)}\|_{*}+\lambda\|Ax-b\|_2^2 &\ \ s.t. X_{ijk}=M_{ijk}\ \ ...
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32 views

Minimization/maximization of system of nonlinear equations

Consider a system of nonlinear equations of the following form: $$F_1(x_2, x_3, x_4...x_n)$$ $$F_2(x_1, x_3, x_4...x_n)$$ $$...$$ $$F_n(x_1, x_2, x_3...x_{n-1})$$ And we wish to simultaneously ...
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nonlinear optimization : restricting search space to “limited” preselected value

My function is nonlinear with respect to a scalar \alpha . However, the calculation of objective function is very time consuming, making optimization also very time consuming. Also, I have to do it ...
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92 views

Dynamic programming at a non linear programming problem

Could you explain to me how we can use dynamic programming in order to solve a non linear programming problem? What do we do for example if we are given the following problem? $$\max (y_1^3-11 ...
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27 views

minmizing frobenius norm($\min_{K \in \mathbb C^{n \times m}} \|KQK^*\|_F$) subject to an equality

In which known approach or algorithm a fat matrix $K\in \mathbb C^{n \times m} ,m>n,$ can be found: $\min_{K \in \mathbb C^{n \times m}} \|KQK^*\|_F$ subject to: $KK^*=I$ $Q\in \mathbb C^{m ...
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30 views

Minimizing summation using Karush–Kuhn–Tucker

Let $a_j, c_j , j=1,...n$ and b be positive constants. $$-$$ Minimize : $\sum_{j=1}^n \frac{c_j}{x_j}$ $$-$$ subject to:$\sum_{j=1}^n a_j x_j =b$ $$-$$ Write down the kkt conditions and solve the ...
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31 views

Optimization by KKT-method

I need to solve the following problem by KKT method. $$ \text{min} \ \ 2xy + 2yz + 2zx \\ \text{subject to} \ x^2 + y^2 ≤ 2, \ 2x + 2y + z = 0 $$ I have gotten as far as setting up the system of ...
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13 views

Using sequence of least squares solutions to solve non-linear problems?

I do know about the iteratively reweighted least-squares and have played around with it to some success finding non-linear solutions (like minimizing non-2-norms to achieve solutions which seem to be ...
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15 views

What is the moment matrix in converting polynomial optimization problem to a quadratic optimization problem

Happy new year, I have a function of the form below \begin{align} f(x,y,z)=\sum_i x_i y_i z_i + g(x)+h(z)\cr x,y,z \in R^n \end{align} where $h,g$ are quadratic functions. My difficulty lies in the ...
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48 views

Linear Optimization Problem with exponential variable

Hey Folks I've encountered an optimization problem which has a linear programming structure but it's coefficients are nonlinear function of another variable. here is the problem: $$\max ...
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69 views

How to efficently solve a convex optimization problem with positive semi-definite Hessian matrix?

Consider the following optimization: $$ f(x)= \min \sum_{i=1}^n \left(x_i-\sum_{j=1}^n x_j\right)^2 $$ Let $g_i(x)=x_i-\sum_{j=1}^n x_j$ , then $$ f(x)= \min \sum_{i=1}^n g_i(x)^2 $$ The ...
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50 views

Positive-Semi-Definite form of Variance?

first thing: I'm an informatics student and know some algebra. However, this seems to be a bit over my head, so please be gentle with me. ;) I have multiple sets of real variables. Let these sets be ...
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75 views

Maximize $ 2^{(-x)} + 2^{(-y)}$ subjected to certain conditions

I am reading through convex optimization and I came across this following problem: \begin{align*} \max \text{ } & 2^{-x}+2^{-y}\\ \text{s.t. } & \frac{1}{1+x}+\frac{1}{1+y}\leq b\\ & ...
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192 views

Monotonic Function Optimization on Convex Constraint Region

So I have the following function, which I want to maximize: $$f(x_1,...,x_n) = \sum_{i=1}^n\alpha_i\sqrt{x_i}$$ (where all $\alpha_i$ are positive), subjected to the following equality and inequality ...
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18 views

finding the least absolute inner product solution for some give vectors

Are there any efficient algorithms (like for PCA) that can find the orthogonal vector of a given bunch of vectors? Mathematically, let $\mathbf{b}_1, ...
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27 views

Is there any approach towards finding all infinite solutions of a set of nonlinear equations when the number of unknowns is more than equations?

I have 3 nonlinear equations with 4 unknowns, with some bound constraints. How can I see if there is a solution to the problem? I wonder if there is a similar approach in nonlinear equations like SVD  ...
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48 views

Maximizing Frobenius norm

I was wondering if anybody has any suggestions on the following problem: Let S be an $n \times n$ real symmetric matrix and $W$ is a real matrix of size $n\times d$; $1\leq d <n$. $$ \text{Find ...
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How to use the Karush–Kuhn–Tucker conditions?

From what I read, the Karush-Kuhn-Tucker conditions are a generalization of the Lagrange Multiplier Method. For the Lagrange Multiplier Method I have been able to find a serie of steps I must do to ...
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29 views

proving that a global maximizer does not exist

Show that no matter what the value of $a$ is chosen, the function $f(x_1,x_2)=x_1^3-3ax_1x_2+x_2^3$ has no global maximizers. Determine the nature of the critical points of this function for all ...
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Proving that a function is coercive

Let $f(x,y)=x^2-2xy+y^2$. I know this is not coercive as along the line $y=x$, when $||x|| ->\infty, f(x,x)=0$. But I don't understand what is wrong with the following way. ...
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36 views

Is $|x^3|$ convex?

Let $f(x)=|x^3|$ on I=$-\infty,+\infty$ Is this convex? How I did was f(x) = \begin{cases} x^3, & \text{if $x>=0$ } \\ -x^3, & \text{if $x<0$ } \end{cases} Then$ f '(x)$ = ...
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34 views

How to Prioritize Constraints of an optimization problem?

I have an optimal control problem of a vehicle the form: $$ \min_{u \in R} \ell (x(j),u(j)) = {\Vert x(j)- x_r(j)\Vert}_Q + {\Vert u(j)- u_r(j)\Vert}_R $$ subject to $$\dot{x}=f(x)$$ ...
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43 views

Are these two optimization problems equal?

The first optimization model: $$ \begin{array}{cl} \arg \min \limits_{C} & \sum\limits_{i=1}^{3}\gamma_i\|{C_{(i)}}\|_*\\ \mathrm{s.t.} & \|A\mathbf{c}-\mathbf{b}\|_2^2+ ...
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18 views

Fitting nonlinear differential equations to correspond a predefined solution

When modeling temporal dynamics of a biological process I stumbled upon a set of differential equations having the following matrix form: ...
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55 views

Levenberg-Marquart with Hessian and gradient

I am minimizing a sum-of-squared-differences function using the Levenberg-Marquardt method. The off-the-shelf numerical implementations I have have looked through (MATLAB, Numerical Recipes in C ...
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3 views

Leveraging the inverse in nonlinear optimization

Consider a non-linear optimisation problem like $$\mathcal{L} = \left\|{\bf x} - f({\bf y})\right\|$$ which we aim to minimise for vector ${\bf y}$ and where $f(.) : \mathbb{R}^N \mapsto ...
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18 views

How find the roots of non-convex function?

How to find a roots of non-convex function f(x)=0, where f is real scalar function of real scalar argument. What methods are exist for it? Or/And where I can to read about it?
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31 views

Minimizing a quadratic function with constraints on some variables

Consider a problem with strictly convex quadratic objective with some of the unconstrained variables. minimize: $x_1^TP_{11}x_1 + 2x_1^TP_{12}x_2 + x_2^TP_{22}x_2$ subject to: $f_i(x_1) \leq0, i = ...
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1answer
101 views

How to find the minimum distance from a point to a set?

Let $M=\{x: x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\le4, x_{1}^{2}-4x_{2}\le0\}$ and $y=(1,0,2)^{T}$. Find the minimum distance from $y$ to $M$, the unique minimizing point and a separating plane. Does anyone ...
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30 views

How to solve the following optimization problem with projection?

How to solve the following optimization problem with projection? \begin{alignat}{1} &\min_{u_+,u_-,s,l\geq 0} \frac{1}{\lambda} \langle A ,(a +u_+-u_-)(a +u_+-u_-)^\mathsf{T} ...
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44 views

Simultaneous diagonalization of two symmetric matrices where one is pd

Let $A,B,C$ be two symmetric $n$ x $n$ matrices, where $B$ is also positive definite. Imitate the procedure to obtain a spectral decomposition of $A$ with respect to $B$, by replacing the constraints ...
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26 views

Examples of complex analysis useful in optimization?

Are there any examples of complex analysis applicable to mathematical optimization problems (preferably non-linear optimization)? I am wondering what advantages the use of complex numbers would have ...
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42 views

How to solve this KKT problem?

Given an optimization as follows: \begin{align} \text{minimize}\quad &c^Tx \\ \text{subject to}\quad &Ax = 0 \\ & \|x\|_2^2 \leq 1 \end{align} where $A \in \Re^{m\times n}$ is of ...
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25 views

What class of problem is a set of equations using inequalities and if-then-else?

Can you please identify what class of problem this is so that I can research algorithms for solving it please? Its a a set of linear equations and inequalities/constraints looking like this: ...
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1answer
32 views

Finding the gradient in least squares

In Linear squares optimization I have A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & \vdots& ...
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25 views

Making a distribution equally up to a maximum … but with a minimum!

So this seems to me like a linear programming problem, but I am getting some odd results. Forgive me, I'm not going to get the terminology correct, I know, so I will present the problem in simple ...
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1answer
31 views

KKT Conditions for NLP [closed]

How may I state the KKT conditions for minimize $f(x) = ax^2$ subject to $Ax \leq b$, $x$ unrestricted?
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25 views

Least squares optimization

In Least Square optimization, A=\begin{pmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^k \\ 1 & t_2 & t_2^2 & \cdots & t_2^k \\ \vdots & \vdots& \vdots ...
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20 views

optimal step length that minimizes the reduced function

Given the problem min $g(y) = y^{T}Ay$ such that $Ay = b$ and $A$ is positive definite. Assume $u$ is the search direction for the reduced function in the reduced space, and $v$ is the corresponding ...
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What is the logic behind the given optimization problem?

I am following a book which has a part on numerical optimization techniques. In order to elaborate Karush-Kuhn-Tucker theorem, they gave the following example: When the unconstrained solution $x=A^+ ...
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27 views

Unique solution of a non-convex OP

I have to find $x$ that minimizes: $$\|x^H\textbf Ax - b\|_2^2 = \sum_{k}(x^H\textbf A_kx - b_k)^2$$ where $A_k$ are 4 x 4 positive definite matrix($A_1, A_2,...A_k$), $x$ is 4 x 1 vector and $b_k$ ...
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scan a box in a circular manner..

I am the moment trying to figure out an expression that is capable of giving my what elements lies in a 2d array, when it is looked at from the center at certain angle out. for instance this is for ...
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33 views

Find $\min \left((x-2)^2+y^2\right)$ $s.t. \>x^2\leq ky^2+1$, $x\geq 0$

Consider the problem $$\min \left((x-2)^2+y^2\right)$$ $$s.t. \>x^2\leq ky^2+1$$ $$x\geq 0$$ where $k \in \mathbb{R}$ is a parameter of the problem. Determine the status of the point $(1,0)$ for ...
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32 views

How to solve following optimization problem?

$$ \begin{eqnarray} & & \min_{X,E} ||X||_* + \lambda ||E||_{\ell 1}\\ & \text{s.t } & \left\{ \begin{split} & x_{ij} \ge 0 \text{ for all the entries of } X \\ & ...
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Nonlinear Optimization Residual Error Calculation with Sign Dependence

I'm working on some code to perform a simple nonlinear optimization. In this scenario, my objective function takes some number of inputs (maybe 3 to 6ish) and will return residuals (maybe 30 or 40). ...
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98 views

How does one choose the step size for steepest descent?

Consider finding the minimal value for any function $g$ from $\mathbb{R}^n$ to $\mathbb{R}$. The method of steepest descent for finding a local minimum for an arbitrary function $g$ from from ...
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44 views

Nonlinear Optimization Problem with nonlinear constraints

How can i solve this problem? $$\max_{a,b,c,d\in\Bbb R} \sqrt{a^2+(3-b)^2}+\sqrt{(b-c)^2+1}+\sqrt{c^2+(1-d)^2}$$ \begin{align*} \text{subject to: }&\qquad 0 \le a \le 1, \quad 1 \le b \le ...
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44 views

KKT Conditions for Euclidean Distances

Suppose that we have an undirected and edge weighted graph $G = (V,E)$. The weight $w_{ij}$ of an edge $\{i,j\} \in E$ determines the Euclidean distance between the vertices $i$ and $j$ s.t. $i,j \in ...
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15 views

Determining Euler-Lagrange equations for a nonsmooth functional

Is there any good resource for understanding how to derive the EL equations for non-smooth problems? In particular, I would like to get them for: $\mathcal{J}(u,v)=\iint_{\Omega} \sqrt{u_x^2 + u_y^2 ...