For questions about optimization and optimization problems with non-linear restraints.

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3
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2answers
40 views

The quadratic case in nonlinear programming

I'm reading about nonlinear programming and I stumbled into the following statement where I started to wonder a bit: Consider the function $$f(\textbf{x}) = ...
0
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1answer
24 views

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 ...
0
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1answer
28 views

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 ...
1
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1answer
35 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 ...
1
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1answer
52 views

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 ...
1
vote
1answer
35 views

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 ...
1
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1answer
27 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 ...
1
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0answers
28 views

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 ...
0
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0answers
20 views

Minimization problem with amplitude constraint

I have the following minimization problem: $$\left\| \bf{A}x - y\right\|^2 \to min $$ $$s.t. \left|x_i\right| < 1, \forall i,$$ where $\bf{A}$ is the complex matrix with size of $(n\times m)$, ...
0
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1answer
27 views

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}): ...
1
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0answers
44 views

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 ...
0
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2answers
56 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 ...
0
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2answers
63 views

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 ...
0
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2answers
45 views

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 ...
1
vote
2answers
71 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 ...
0
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0answers
38 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 ...
0
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0answers
26 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 ...
0
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1answer
21 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 ...
1
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0answers
25 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.
0
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0answers
17 views

What is a minimal equation system?

In the optimization seminar I have to study the quadratic linear ordering problem. And there is one lemma saying some equations form a 'minimal equation system' of a polytope. Does anybody know, what ...
2
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1answer
111 views

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 ...
0
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1answer
39 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 ...
0
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1answer
342 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 ...
1
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0answers
34 views

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 $d < p$. ...
0
votes
2answers
28 views

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 ...
1
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0answers
44 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 ...
0
votes
1answer
41 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 ...
1
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0answers
28 views

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 ...
2
votes
1answer
30 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. ...
1
vote
2answers
50 views

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 ...
1
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0answers
27 views

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 $\mu$, ...
0
votes
1answer
41 views

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 ...
0
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2answers
56 views

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 ...
0
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0answers
35 views

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 ...
0
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0answers
96 views

Maximize the maximum Eigenvalue under a diagonally constrained matrix

Suppose we have $N\times N$ Hermitian matrix $\mathbf{A}$ I want to find the real $N\times N$ diagonal matrix $\mathbf{D}$ that maximizes the sum of the maximum Eigenvalues : $\mathbf{D}=\arg\max ...
2
votes
2answers
86 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. ...
0
votes
0answers
21 views

min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
2
votes
1answer
54 views

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: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
3
votes
1answer
66 views

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; ...
1
vote
1answer
45 views

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 ...
0
votes
1answer
26 views

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 ...
0
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0answers
19 views

Monotonic transformation in numerical optimization

Taking the logarithm of the Cobb-Douglass utility function ($u = x_1^a * x_2^b$) yields a utility function whose argmin is somewhat easier to derive. Since the logarithm is monotonic for $u>0$, we ...
2
votes
1answer
49 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\\ ...
2
votes
0answers
55 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 ...
0
votes
1answer
49 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 ...
0
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0answers
19 views

Constraint approximation in non-linear optimization

In given non-linear optimization problem \begin{equation*} \begin{aligned} & \underset{x \in\mathbb R^n}{\text{maximize}} & & f(x) = \alpha^2 \\ & \text{subject to} & c(p(x)) \le ...
3
votes
1answer
37 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 ...
0
votes
1answer
32 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) ...
1
vote
1answer
63 views

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 ...