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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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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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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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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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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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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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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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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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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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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28 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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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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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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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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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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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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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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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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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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42 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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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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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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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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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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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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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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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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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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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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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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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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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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(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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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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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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90 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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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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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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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 ...
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Find all the points satisfying the Fritz John conditions

Consider the problem $$\min \>x^2+y^2 $$ $$s.t.\> x^2-(y-1)^3=0$$ Find all the points satisfying the Fritz John conditions Solution The FJ conditions are $$2x+\mu_1 2x=0$$ $$2y-\mu_1 ...
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A nonlinear optimization problem with difficult Kuhn-Tucker system of equations

I know about the sufficient optimality theorem Kuhn-Tucker, and this problem can use the Kuhn-Tucker theorem directly, but ridiculously, I got stuck on the system of equations to find one root for ...
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how does sequential axis search work?

I see that some algorithms that need to search for a global minimum in multiple dimension space, say find x and y to minimize f(x,y), instead of searching in x,y simultaniously, starting from initial ...
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What are the general algorithm and precise mathematical language that can optimise the nodes in a graph?

Recently I came across this via social media Out of curiosity (and because I am a visual learner) using the paragraph in the article, I end up drawing some kind of mixed graph, as shown I then ...
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Linear Matrix Inequality - “HOW TO”

I have a tough time understanding how to use Linear matrix Inequality to solve simple inequality problems. I would appreciate a simple "How to" on the following examples. $$ \bar PA + A \bar P - ...
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Non-binding constraints with positive shadow prices (matlab)

The output of fmincon indicates positive shadow price for linear constraints, although the corresponding constraints are not binding. What could be wrong mathematically? I've checked the code but ...
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40 views

Transportation problems

i'm a master student at the deparment of statistics. And i will prepare a presentation on transportation problems in the course of optimization (or linear programming / mathematical programming) I ...
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58 views

Weakly unimodal function using Golden Section Search

I was going through the Golden Section Search https://en.wikipedia.org/wiki/Golden_section_search and as I understand it should work for every unimodal function. Here, the definition of unimodal ...
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Nonlinear Programming

I have the following non-linear programmig problem that I have arrived at after various manipulations. I have to find the set of values for $x$ and $y$ that satisfy the following: $$ x^{n}+y^{m}=C $$ ...