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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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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60 views

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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24 views

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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31 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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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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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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17 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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29 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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42 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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41 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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24 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
30 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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1answer
23 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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11 views

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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1answer
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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54 views

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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94 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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40 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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13 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 ...
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13 views

Minimizing the non-linear cost of successive Bernoulli trials

Suppose that performing a Bernoulli trial with probability $p$ costs $$ \gamma(p) = \ell^2 \, \tan\left(\frac{\pi}{2} \, p\right) + 8 \ell $$ for a fixed positive integer $\ell$. Furthermore, fix $0 ...
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12 views

Nonlinear optimization within a critical subspace

I have a constrained optimization problem of the following nature: $$x^\ast=\left\{ \begin{array}{rl} \arg\min_{x\in\mathbb{R}^m} &E_1(x)+E_2(x)\\ \mathrm{s.t.}& c_i(x)=0\ \textrm{for}\ 0 \leq ...
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27 views

How to express non-linear equation

I asked this question over on tex.stackexchange.com: http://tex.stackexchange.com/questions/280360/how-to-express-non-linear-equation-in-latex?noredirect=1#comment675446_280360 they suggested that I ...
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11 views

Motivation of acute angle principle

Acute angle principle: Let $\Omega$ is open subset of $\mathbb{R}^n$,$\theta\in \Omega$ . $f:\overline\Omega\rightarrow\mathbb{R}^n$ is continuous.And $\forall x\in\partial \Omega,(f(x),x)\ge0$,then ...
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1answer
38 views

How to solve/deal with the following optimization issue?

I got the objective function $\displaystyle f(\alpha)=\alpha\cdot \left(1-\frac{\binom{N+K}{K}\beta^K}{\sum\limits_{k=0}^{K}\binom{N+k}{k}\beta^k}\right)$, where $N$ and $K$ are positive integers, ...
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17 views

Validity of nonlinear optimization with exponential type inequality constraint as KKT / Lagrange multipliers?

Given positive coefficients $h_i, \beta_i$ and $k$ we have the minimization problem $$\displaystyle \min\sum_{i = 1}^n h_{i}s_i\\ \text{subject to} \displaystyle\sum_{i = 1}^n \alpha^{s_i}\beta_i \leq ...
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1answer
33 views

Generalized Farkas Lemma

Farkas lemma can be stated as follow: If for all $\mu$ such that $\mu^T\cdot a_i \geq 0$ implies that $\mu^T\cdot b \geq 0$ then $b=\sum \lambda_i a_i$ with $\lambda_i \geq 0$ I need a generalized ...
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16 views

Converting an optimisation problem to an integer linear formulation

Is there a way to convert the following to a linear formulation? In other words, is there a workaround for the absolute value in the objective function? Minimise: ...
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25 views

How to linearize a constraint of the form of a product?

Is there a way to linearize a constraint of the form: $$\prod\limits_{ i=1 }^{ n }y_i\geqslant b,$$ where $y_i$ are discrete variables in the set $\{1,2,\ldots,2^m\}$ for some $m>2$ and $b$ is a ...
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1answer
58 views

Min problem by using Lagrange method

$$\min x^2+y^2 $$ $$\text{s.t.}\ \ (x-2)^2+(y-3)^2\le 4 \ \ \ \text{and} \ \ \ x^2=4y$$ Please explicitly solve this question by using Lagrange multiplier method. I accept $(x-2)^2+(y-3)^2=4$ ...
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21 views

(Convex) reformulation of a nonlinear program

Consider the following program: \begin{eqnarray*} \min_{\mathrm x}\sum_{i=1}^{n}{\sum_{j=1}^{n}{\big(x_i(Sx)_i-x_j(Sx)_j\big)^2}}\\ \mathrm{subject\; to}\quad \sum_{i=1}^{n}{x_i}=1 \\ x_i\geq 0 ...
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1answer
69 views

What is the difference between min and max constraint problems?

For example, let's consider these two min max optimization questions (1) $$\max \ \ g(x,y)=xy$$ $$ \text{s.t. } x^2+y^2=1$$ (2) $$\min \ \ g(x,y)=xy$$ $$ \text{s.t. } x^2+y^2=1$$ Solution: By ...
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21 views

Condition for stationary point without maxima or minima

Consider $f(x) = \frac{1}{2}x^{T}Qx - c^{T}x$. Under what conditions on $Q$ does $f$ have a stationary point, but no local maxima or minima? I need help refining my thoughts here. I don't think I ...
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96 views

Binary integer program with nonlinear function

I have given a matrix $A^{m \times n}$ and I am looking for a submatrix $B^{m \times k}$ for a given $k$ that maximizes the following expression: $$\sum_{i=1}^m \max_{j \in \{1 \dots k\}} B_{i,j}$$ ...
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1answer
58 views

Prove or disprove convexity

I am dealing with the following function $f:\mathcal{R}^n \rightarrow \mathcal{R}$, how can I prove or disprove the convexity of the following function? $$f(x)=\|x-\frac{Ax}{\langle x,b\rangle}\|_2$$ ...
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18 views

Is there any “equivalence” to maximizing $\inf{f_i(y)}$?

I have to maximize the function $g(y) = inf_i{\|y - x_i\|_2}$ subject to $y\in B_0(1)\subset\mathbb{R}^n$. Then I thought that maybe there is an averaging or mollifying of the functions (using ...
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1answer
25 views

Solve the above program [closed]

Consider the problem of covering the triangle with vertices at the points $(0, 0), (0, 1),$ and $(1, 0)$ with a ball of smallest radius. $$\min r$$ $$s. t. \> x ^2 + y ^2 ≤ r$$ $$(x − 1)^ 2 + y ^2 ...