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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Goldstein test in nonlinear programming

I'm reading about nonlinear programming and the Goldstein test. Here is the definition from my book: A line search accuracy test that is frequently used is the Goldstein test. A value of ...
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Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...
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73 views

Can Gomory's cutting plane be used to solve Mixed Integer Linear Programs?

Do you know if Gomory's cut can be used to solve MILP problems ? I have read that Gomory's cut is useful when all variables are integer, but what if just some of them are integer ? Is is necessary ...
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Matrix Maximization

I would like to solve the following optimization problem for a matrix $X$ which is symmetric and positive-semidefinite: $$ \mathrm{maximize} \, \, \, f(X) = \log \mathrm{det} X - k_1 \log(k_2 + a^T X ...
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Order of convergence for the method of false position

I'm reading about the order of convergence of the method of false position and there is one tricky point in the proof I don't understand. The method itself for finding the minimum $x^*$ of a function ...
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59 views

Maximizing the probability of multivariate hypergeometric distribution

Suppose there are $R$ number of red balls, $B$ number of black balls, and $W$ number of white balls in an urn. The total number of balls are $N$, thus, $N = R + B + W$. I draw $M$ number of balls from ...
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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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Optimality conditions in convex programming

I'm reading about Zero-order conditions in Nonlinear Programming and the following confuses me (my questions are below the theory): Consider the set $\Gamma \subset E^{n+1} = \{(r,\textbf{x}): ...
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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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71 views

Nonlinear Optimization problem

Function $f(x) \in \mathbb{R}^n$, $(n\geq 1)$, depend on one parameter $x \in \mathbb{R}$. Performing a nonlinear transformation of $f(x)$, we obtain function $g(y) \in \mathbb{R}^n$. This ...
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Optimizing elementary symmetric polynomial on the unit sphere

I'd like to optimize $x_1 x_2 x_3 + x_1 x_2 x_4 + x_1 x_3 x_4 + x_2 x_3 x_4$ on the unit 4-sphere. I'm thinking I should do lagrangian optimization, but I'm having trouble solving the resulting ...
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optimization problem: finding an hyperplane separating one point from a set of pointy maximizing the distance

I have this problem: I have a set of n-dimensional points $P$. I have one more n-dimensional point $q$. The points in $P$ are linearly separable from $q$ (i.e. it always exists an hyperplane $n^t x ...
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105 views

Forbidden range for a linear programming variable

I would like to express a linear program having a variable that can only be greater or equal than a constant $c$ or equal to $0$. The range $]0; c[$ being unallowed. Do you know a way to express this ...
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104 views

How to characterise this non-linear optimisation (linear objective function, non-linear constraints)

I was wondering if someone may be able to help me characterise this optimisation problem as I am struggling to find a numerical library that will solve it and I suspect it is because I am using the ...
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66 views

Fitting noise with Levenberg–Marquardt algorithm

I've got a sample of noise from a microphone, and I'm trying to fit a curve to the data using the Levenberg–Marquardt algorithm. However, I can't seem to find a good starting function. I've tried a ...
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24 views

Is it possible to approximate or bound this nonlinear mapping L h(x)?

I wanted to make sure whether the following approximation is possible. I have a vector ${\bf{x}} \in \mathbb{R}^N$ where the elements of the vector $\bf{x}$ are random variables. There is a matrix ...
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57 views

How to solve Bellman's optimal equation from the first principle

How to solve the following set (finite) of equations $$ v_*(s) = \max_{a\in A(s)} \sum_{s'} p(s'|s,a) [r(s,a,s') + \gamma v_*(s')]$$ $p$ and $r$ functions are given.
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Linearization of a product of two decision variables

I am trying to solve a problem that involves constraints in which products of two decision variables appear. So far, I read that such products can be reformulated to a difference of two quadratic ...
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96 views

prove that a black-box multivariable problem is convex or concave

First of all I am not mathematician. I want to solve a very complex black box function with several constraints having to do with electrical power flows in electrical grids. At the moment I use the ...
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2k views

Newton-Raphson Method for Non-linear System of 3 variables in Matlab

I am trying to solve 3 non-linear system of 3 variables using the newton-raphson method in matlab. Here are the 3 non-linear equations: \begin{equation} c[\alpha I+ k_f+k_d+k_ns+k_p(1-q)]-I \alpha =0 ...
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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 $d < p$. ...
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Determining the minimum value

Function $W(t,x)$ is defined as \begin{equation} W(t,x)=\sum_{i}\alpha_i e^{-\beta_i(t-x)}, \end{equation} where $\alpha_i$ is real and $\beta_i$ is real and positive, Then $\Psi$ is defined as ...
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48 views

Maximize the expected values of a function with constrain

Consider $p_1,p_2,...p_N$ are probabilities arranged in ascending order. $n_1, n_2,...n_N$ are numbers which are arranged in geometric progression. I want to Maximize E= $\sum\limits_{i=1}^N p_i\cdot ...
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151 views

Is there any theoretical upper bound on the second derivative of a twice-differentiable function?

Lets assume that f(x) is a twice-differentiable and nonlinear function, where x is bounded by the interval l ≤ x ≤ u, and the function itself is bounded by L ≤ f ≤ U. We know the values of l, u, L and ...
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First order necessary conditions for nondifferentiable nonconvex minimization problem

I am interested in first order necessary conditions for the following minimization problem where the function $f$ is continuous, nondecreasing and concave, with $f(0)=0$, but not necessarily ...
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58 views

How to prevent a convex optimization from being unbounded?

I'm novice in optimization and have a convex optimization function of form $\sum_{i,k} p_{k,i}*\log{p_{k,i}} $ to minimize with the following constraints: $\forall i, a_i = \sum_{k=1}^{m} b_k. ...
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Why are the KKT conditions sufficient in this case?

I need to solve this problem: minimize $f(x,y)=5x-xy-50+10y$ subject to: $g_1(x,y)=18-xy\leq0$ $g_2(x,y)=x+y-11\leq0$ $x,y\geq0$ I found the optimal solution: $(x,y)=(6,3)$ But the Hessian ...
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Solve Van der Pol equation by Padé approximation

I want to solve the Van der Pol equation: $$f''+ \mu \, (f^2-1)f'+f=0, \quad f = f(t),$$ by Padé approximation. I know the solution should be the combination of $\sin{t}$ and multiplied by $\mu$, ...
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Is the optimization of the following composite function even possible, and if so, how would I go about solving it?

Hi Guys, So when I formulate a problem I am trying to solve for work, the above (please see attached figure) optimization framework results. I am not too familiar with optimization techniques ...
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Regularization vs. Inequality Constraint

For what values of a regularization parameter $\alpha$, there is an equivalent inequality constraint in convex optimization? In particular, in the convex optimization problems below $$ \text{ Problem ...
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Finding a solution using the principle of maximum entropy?

I have set of linear constraints and would like to find an answer to its unknown variables, $p_i$'s. One of my options to find a solution for $p_i$'s using maximum entropy problem, $\max(\sum - p_i ...
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372 views

How to maximize an entropy function?

I'm very novice in optimization and have a convex optimization function of form $\sum_{i,k} p_{k,i}*\log{p_{k,i}} $ to minimize with the following constraints: $\forall i, a_i = \sum_{k=1}^{m} b_k. ...
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Can this multidimensional non-linear equation with constraints be minimized analytically?

I wish to find the vector of real numbers, $\mathbf{w}$, that minimizes the function: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
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Are all non-convex problems created equal?

The distinction between convex and non-convex problems is usually dubbed as the distinction between easy and hard problems. While in the convex case you are golden (local optima are global optima; ...
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Optimal String Shape Problem

So here is the problem I am working on, Given a curve of length L connecting the points (0,1) and (1,0) find an expression for the equation of the curve that minimizes the area underneath it. In ...
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Can $\min f'x$ s.t. $(a'x - b)^2 \le d $ be written as a SOCP?

It does not appear to be significantly different from the form listed here: http://en.wikipedia.org/wiki/Second-order_cone_programming with (in article notation) $i = {1}$, $ A = a$, and $b$, $d$ as ...
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115 views

Least squares problem with orthonormality constraints

Given $y_1,\ldots,y_n\in \mathbb{R}$,$w\in \mathbb{R}^d$, and $x_1,\ldots x_n\in \mathbb{R}^D$, how do we solve the following optimization problem \begin{align} \min_A \sum_{i=1}^n (y_i-w^TA^Tx_i)^2\\ ...
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301 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 ...
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63 views

Optimization problem with a minimization sub-problem as a constraint

I have a problem, for predefined $x_0,z\in\mathbb{R}$, which looks like $$\min_{\alpha,x} \sum_{i=1}^n \alpha_i f_i(x_i,z) $$ subject to \begin{align} \sum_{i=1}^n \alpha_i &= 1 \\ \sum_{i=1}^n ...
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49 views

How to reformulate this Set covering problem?

I am trying to solve the following implementation of the set covering problem of a crew rostering problem. Here constraint (19), meant to create a 12-hour break between the different shifts taken by ...
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52 views

Implement $\max$ with a closed form expression?

I have 2 functions: $f(x)$ and $g(x)$. Both of them range in $[0,1]$. Is there some way to define a $h(x)$ that efficiently takes the greater one of $h(x)$ and $g(x)$, i.e. $h(x) = \max \{ f(x),g(x) ...
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Showing a function is concave

Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$. Since we have ...
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Identifying saddle points of a constrained nonlinear function with three variables

I know that if the Hessian matrix of a multivariable function at a given stationary point has both positive and negative eigenvalues then that stationary point must be a saddle point. Does the same ...
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Comparing the hardness of optimizing two similar, but different expressions

Suppose we have binary variables $y_1, ..., y_n$. To make the representation simple, we show the concatenated vector as $\mathbf{y} = (y_1, ..., y_n)$. Consider the two following functions: $$ ...
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Solving nonlinear matrix inequality - transformation to LMI

I have a nonlinear matrix inequality problem where $A,B,C$ and $M$ are known and T is unknown and I would like to find $T$ that satisfies $\begin{bmatrix} T^T M T + A & B \\ B^T & ...
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Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
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QR-Decomposition of matrix valued function

I already posted the following question on MO, but id did not raise much interest there. Maybe the title is too elementary to gain research interest. Suppose I have a matrix valued function $$ ...
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What is some prerequisite to study nonlinear programming?

What is some prerequisite to study nonlinear programming? I already know calculus and linear programming is two perquisite, what else?
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Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...