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5
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0answers
194 views

I need help about some compactness arguments

I need help to find some compact sets for my engineering problem. Through this page I learned quite much about it however since I have neither read a book yet nor have an experience I am not able to ...
4
votes
0answers
27 views

Nonlinear optimisation of Expectation

I am preparing for my exams and I can't get my head around the following question. I know there exists a general method for solving these problems but I don't know where to start. I would greatly ...
4
votes
0answers
134 views

Why Compactness is Necessary at Minimax Theorem

According to Von Neumann's minimax theorem, I have $$\max_{x\in X} \min_{y\in Y}f(x,y)=\min_{y\in Y} \max_{x\in X} f(x,y)$$ for some compact sets $X$ and $Y$ and a convex (in $y$), concave (in $x$) ...
4
votes
0answers
295 views

(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
3
votes
0answers
27 views

Solve non-linear equation with Matrices

I'm looking more for hints than specific answers, although I would be extremely grateful if provided with one. The problem I have is as follows: $$ -\Sigma (A+\Lambda_1)+I=0 $$ Here A is a constant, ...
3
votes
0answers
38 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
3
votes
0answers
23 views

How does this polar function behave?

I came across this question in my textbook for Nonlinear Optimisation and I don't know what to do: Consider the function: $$ f(x_1,x_2)=(r-1)^2-\frac{1}{2}(r-1)^2\cos \left( \frac{1}{r-1}-\phi ...
3
votes
0answers
67 views

Solver for sparse linearly-constrained non-linear least-squares

Reposted from stackoverflow on the advice of Nick Rosencrantz: Are there any algorithms or solvers for solving non-linear least-squares problems where the jacobian is known to always be sparse, and ...
3
votes
0answers
264 views

Optimizing non linear programs of two variables

The scenario is; We've got $n$ stationary 360$^{\circ}$ sensors in an confined area (each sensor is located at some arbitrary $\left(x,y\right) = \left(x_{n},y_{n}\right)$), once a unit $t$ enters ...
2
votes
0answers
64 views

Finding optimal velocity profile using Dynamic Programming

This question has been asked on scicomp but I thought maybe it's more a mathematical problem of how Bellman's idea is to be applied here. The main problem for me is: How to introduce the time ...
2
votes
0answers
24 views

Levenberg's original article “A method for the solution of certain problems in least squares”

Does there exist any digital copy of the original article (or a transcript) K. Levenberg, A method for the solution of certain problems in least-squares, Quart. Appl. Math. 2 (1944): 164-168? It is ...
2
votes
0answers
41 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 ...
2
votes
0answers
79 views

Maximize a function subject to the constraint $x^2+y^2=R^2$

Please help me how to deal with maximization of function $$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$ subject to the constraint $x^2+y^2=R^2$. Using Lagrange ...
2
votes
0answers
35 views

Calculating second derivative of $g(\alpha) = f(\textbf{y}(\alpha))$

I'm having problems with the second derivative of the function $g(\alpha) = f(\textbf{y}(\alpha))$ (which I will define more precisely below). I tried calculating it myself, could anyone just simply ...
2
votes
0answers
47 views

the objective function $\|F\|_F^2$ is quasiconvex in the optimization?why?

I have read a paper, but I can not understand one optimization thoroughly.Generally, Frobenius norm of one matrix, $\|F\|_F^2$, as the objective function is convex, so we can resolve it not using the ...
2
votes
0answers
71 views

Steepest Descent/Newton

Suppose these over-determined system of equations: $$ |\mathbf{x}^T\mathbf{v_n}| = A, \qquad n = 1,2,\cdots,N-1 $$ $$ \mathbf{v_n}= [1 \quad w^n \quad w^{2n} \quad \cdots \quad w^{(N-1)n}]^T , ...
2
votes
0answers
68 views

System of many non-linear (quadratic) first order O.D.E. (numerical strategy or simplification)

I have a large system (N>100) of equations $\frac{d\vec{P}}{dt}= A(t) + B(t) \vec{P} + \vec{P}^T C(t) \vec{P}$ where $\vec{P}$ is a vector of N functions of the variable t. What is the correct ...
2
votes
0answers
65 views

Nonlinear optimization of constraint parameter - subdifferential?

Disclaimer: I discovered that the FAQ suggests to post research-level to mathoverflow instead of math.stackexchange. I "moved" the question accordingly, cp. post at mathoverflow. Sorry for the ...
2
votes
0answers
276 views

SDP relaxation of non-convex QCQP and duality gap

Short version Is there a duality gap between a QCQP problem and the SDP problem obtained through lagrangian relaxation? A paper I'm studying is using this fact, but I cannot achieve the authors' ...
2
votes
0answers
53 views

Linearization of a dynamic system, $\small 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $

I have a dynamic system, with in signal u and out signal y. The system is described with the following differential equation $$ 10 \frac{d^2 y}{dt^2} + \frac{d y}{dt} = u^2(t) $$ The task is to ...
2
votes
0answers
32 views

Convex formulation of a nearly convex optimization problem

The following problem has come up in my studies of logarithmic norms. I wish to find $\mu \in \mathbb{R}$ and a positive semidefinite $B$ so as to minimize the convex function $c \mu - \log\det(B)$ ...
2
votes
0answers
218 views

sufficient condition for KKT problems

For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that: "The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
2
votes
0answers
176 views

Optimization over function spaces

There are many methods to solve optimization over standard spaces such as real and complex numbers or vector spaces of these. In my case I have a vector space of functions of a real value to find. ...
2
votes
0answers
591 views

Convex minimization over the Unit Simplex

I have a simple (few variables), continuous, twice differentiable convex function that I wish to minimize over the unit simplex. In other words, $\min. f(\mathbf{x})$, $\text{s.t. } \mathbf{0} \preceq ...
1
vote
0answers
18 views

Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
1
vote
0answers
11 views

Maximizing the “uniformity” of a distribution subject to moment constraints

I want to develop a continuous probability density, subject to moment constraints, that is maximally "uniform". A maximally uniform density is a density that has the smallest maximum probability ...
1
vote
0answers
27 views

Gradient descent via polynomial approximation

It seems that most proofs of convergence for gradient descent algorithms rely on strong conditions on the first and second derivatives of the function, for instance that $$|f''(x)| \leq K$$ over the ...
1
vote
0answers
26 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
1
vote
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 ...
1
vote
0answers
34 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 ...
1
vote
0answers
24 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.
1
vote
0answers
32 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 $d < p$. ...
1
vote
0answers
43 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 ...
1
vote
0answers
24 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 ...
1
vote
0answers
26 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 $\mu$, ...
1
vote
0answers
31 views

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 ...
1
vote
0answers
17 views

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: $$ ...
1
vote
0answers
38 views

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 & ...
1
vote
0answers
13 views

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?
1
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0answers
27 views

Inequality optimization, KKT condition.

So we have the problem: maximize $x^2+y^2$ subject to $x^2-y \leq3$ and $y\leq 1$. And I sorted out the KKT conditions for the problem (is here where the problem is?): $2x=\lambda _12x$, ...
1
vote
0answers
14 views

General 2D taylor surfaces from axial behaviour and discrete points

I have a problem as follows: I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete ...
1
vote
0answers
62 views

using lsqcurvefit to fit piece-wise linear

I would like to use this function to fit piece-wise linearly to a set of data. Namely, I want to fit them with several linear segments. Including other requirements, I would not want the segments ...
1
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0answers
69 views

transforming nonlinear matrix inequality to LMI

I faced some nonlinearity in my problems. I need to check a matrix inequality condition in order to check the feasibility of designed controller through a continuous design problem. My problem is that ...
1
vote
0answers
44 views

How to minimise an objective function which is not a direct function of the decision variable?

I have a problem with partitioning a water network by closing some pipes. I use some graph theory techniques to find some candidate pipes to close; but to select which pipes among them to close (my ...
1
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0answers
32 views

Effect of approximating a non-differentiable function on optimisation of minimisation

I am looking at a problem of constrained minimization, where the function to be minimized contains the Heaviside function, and as such is not twice continuously differentiable. My question is what ...
1
vote
0answers
31 views

Representing a 2D function as a sum of rectangles of arbitrary shape and orientation

Suppose I am given a non-negative function $f(x,y)$ defined for $x \in [0,1]$ and $y \in [0,1]$. I'd like to represent this function as a weighted sum $w_i$ of a small number of rectangular apertures. ...
1
vote
0answers
505 views

Significant improvement when I use lsqnonlin function with wrong sized X

I was trying to solve a nonlinear least-square optimization problem using matlab function lsqnonlin with default algorithm trust-region-reflective. Let the optimization problem be "minimize ...
1
vote
0answers
54 views

Levenberg-Marquardt, QR decomposition

Could anybody explain, how the Levenberg-Marquardt method may be solved using the QR decomposition? I know a current solution ...
1
vote
0answers
228 views

How to solve nonlinear constrained optimization in Matlab?

I have to solve a nonlinear constrained function in matlab, and I am not familiar with it's commands. the problem is: minimize $E(b,c)$ constraints: $k1< c\sqrt{b}< k2 ; c/6>k3$ Note: E(b,c) ...
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0answers
55 views

Some problems with a proof of the Farkas Lemma

The following is a proof of the Farkas Lemma that is creating me quite some problems. [I presented the all proof simply to point out the notation used by the author.] My problem is with the last part ...