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1
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5answers
236 views

Independence of Rotation Matrix Definitions

I am trying to solve a system of non-linear equations. I know that 9 of my variables put together form a 3x3 rotation matrix $$ A = \left( \begin{matrix} a_{11}& a_{12}& a_{13}\\ a_{21}& ...
2
votes
2answers
586 views

Lasso with linear constraints

I want to efficiently solve the following optimization problem: \begin{align} \min &\quad \left\|\mathbf{x}-\mathbf{x}_0\right\|_2^2 + \lambda\left\|\mathbf{x}\right\|_1\\ \text{Subject to}& ...
1
vote
1answer
221 views

Optimization of multivariate non-linear function with linear constraint

I have this huge and ugly function. $$ f\left(x,y\right)= $$ $$ ...
1
vote
0answers
46 views

equivalence between primitive and dual

I have a problem about the duality gap of the primitive problem and the dual problem. This problem comes from a probabilistic model named Lagrangian UVM. As this figure shows, this is a trinomial ...
0
votes
0answers
109 views

Solving for Analytical Maximum Likelihood Estimate Parameters

I have a statistical model whose parameters I would like to find a closed form maximum likelihood estimate for if possible. There are two parameters and I can solve for one, but the second is a bit ...
0
votes
1answer
277 views

Generating a random monotonically increasing polynomial?

Given a polynomial $y : \mathbb{R} \mapsto \mathbb{R}$ of degree $p$: $$ y(x) = \sum_{k=0}^p c_k\, x^k,$$ can a random set of coefficients $\{c_0, \cdots ,c_p\}$ be generated such that $y$ is ...
0
votes
0answers
307 views

Bivariate function maximization

I have a bivariate function like $ f(x,y) = \frac{1}{x^3 \sqrt{\pi}}. e^{\frac{2-x}{x^2}} . y^3 . e^{3.y \over 3-y} $ and I want to find its global maximum over a range of $ x \in [0, 200] \text{, ...
1
vote
0answers
132 views

optimization of polynomial function with linear constraints

I have a polynomial function, which is actually the Cobb-Douglas production function, of the form $f(x,y) = \frac{\{x^{\alpha} y^{1-\alpha}\}^{1-\gamma}}{1-\gamma}$ with linear constraint K(x,y)= ...
0
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1answer
46 views

Degeneracy of the analytic center of a set of linear inequalities

I have a question about the degeneracy of the analytic center of a set of linear inequalities. When the set of linear inequalities is degenerate, I guess that the analytic center would also be ...
0
votes
1answer
41 views

Discontinuous optimizer but continuous optimal

Consider a locally-bounded, continuous, positive-semidefinite function $f: X \times Y \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact, $Y \subseteq \mathbb{R}^m$. For each ...
1
vote
1answer
111 views

Question regarding Kuhn-Tucker multiplier

I have a problem which I am unable to solve. If we consider the following problem $\min f(x)$, $G(x) = b$; where $f$ is in $C^2(R^n)$, and $G$ from $R^n$ to $R^m$ is a $C^2$-function, $G = ...
1
vote
0answers
235 views

A convex programming problem involving sum of logarithms of linear functions

Here is a convex programming problem I encountered while working on an estimation problem for a mixture of multinomial distributions. We have a matrix $A_{m \times n}$ containing non-negative real ...
3
votes
2answers
403 views

Multilinear optimization

Are there any efficient algorithms to solve, multi-linear objective and multi-linear constraint optimization problems? The multilinear functions are sums of bilinear, trilinear (and so on) terms ...
1
vote
2answers
138 views

Is nonlinear conjugate gradient a quasi-newton optimization technique?

Can the non-linear conjugate gradient optimization method with Polak-Ribier line-search choice, be named as a quasi-Newton optimization technique? If not, why?
1
vote
1answer
732 views

Set of symmetric positive semidefinite matrices is a full dimensional convex cone.

If $S_n^+$ is the set of all symmetric positive semidefinite $n \times n$ matrices with entries in $\mathbb{R}$, how does it follow that it is a full dimensional closed convex cone in ...
0
votes
0answers
273 views

Can this non-linear optimisation problem be converted to a linear?

I have to minimize the function: $F(x)$ $F(x) = \sum_{i=1}^{M}||x_{i+1} - x_i - K(\frac{x_{i+1} + x_i}{2})||^2 + ||x_1-c_1||^2 + ||x_N-c_2||^2$ , where $x$ is a vector of $N$ scalars, $c$ are ...
1
vote
0answers
62 views

Nonlinear Optimization in Probability

Consider a locally-bounded function $f: X \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact and $Z \subseteq \mathbb{R}^m$ is closed. Let $m(\cdot)$ be a measure on ...
3
votes
3answers
268 views

simple-looking non-convex optimization problem

I want to solve the following problem: Maximize $\sum_{i=1}^n\log(1+\lambda_i^2)$ subject to $\lambda_i >0$ and $\sum_{i=1}^n\lambda_i = M$. I was wondering how I could cast it as a convex problem. ...
0
votes
0answers
140 views

Matlab: Solving $u_t = f(u) u'' + g(u) u' + h(u) u$

I am trying to solve this equation (numerically) $$\dfrac{\partial u}{\partial t}=\dfrac 3 x\dfrac \partial{\partial x}\left(\sqrt x \frac \partial {\partial x}(\nu(u,u^2,\cdots,u^k)u\sqrt ...
3
votes
2answers
155 views

Maximum of a product of a polynomial with positive coefficients and a finite sum of exponentials with negative coefficients on $[0,+\infty)$

Prove or disprove that $$ f(x)=\left(\sum_i a_i x^i\right)\left(\sum_j b_j e^{-\lambda_j x}\right) $$ where $\forall i, a_i>0$, $\forall j, b_j>0,\lambda_j>0$, and both sums are finite, ...
2
votes
2answers
1k views

Algorithm for GRG2 method of solving non-linear least square

I have been looking for quite a while for an algorithm for the GRG2 method either in a .net assembly or an algorithm i could program myself but I cant find a decent representation of the algorithm to ...
1
vote
1answer
146 views

Fixed point stability of piecewise linear system

I have an autonomous system of nonlinear equations of the form: $$Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0$$ where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. ...
1
vote
1answer
83 views

Least squares and (non-)linearity of parameters

I have a question about least squares and about what happens, if the function that we minimize, $E(P)$, is not linear in its parameters $P$. Assume we want to minimize a function (the exact terms are ...
0
votes
1answer
212 views

How to optimize entropy under linear constraints?

My problem is quite cumbersome. In general, it can be modelled as a non-linear programming problem, with linear constraints and non-linear objective function. The objective function is conditional ...
0
votes
1answer
105 views

How to optimize nonlinear goal under linear constraints?

I have a "linear" equation set as follows, with nonlinear optimization goal. P(0) + P(1) = 1 P(0, 0) + P(0,1) = P(0) P(0) < 1 P(1) < 1 P(0,0) > 0 P(0,1) > 0 ...
0
votes
1answer
699 views

How do I construct the Jacobian for use in a Levenberg-Marquardt algorithm.

I am working on a 3D reconstruction system and I am looking to use a Levenberg-marquardt algorithm to do bundle adjustment. I am not too sure about how LM works and what it requires. The model I am ...
7
votes
1answer
280 views

Minimization of $\sum \frac{1}{n_k}\ln n_k >1 $ subject to $\sum \frac{1}{n_k}\simeq 1$

Looking at an algorithm for minimizing $\sum_{k=1}^{m} \frac{1}{n_k}\ln n_k > 1$ subject to $\sum_{k=1}^{m}\frac{1}{n_k} = 1$ in which $n_k$ are positive and in general non-sequential integers, I ...
1
vote
0answers
162 views

Optimization via Simulation

I want to minimize and objective function $\hat{B_i}$ $i\in l$, which can be computed by a matlab code (assume $\operatorname{findB}(a, b, c)$ returns $B$. I have the following optimization problem: ...
0
votes
1answer
1k views

Differences behind different methods of fminunc in MATLAB?

Assume I have some .m file with a function (and it's gradient) to be used by fminunc() in MATLAB for some unconstrained optimization problem. To solve the problem ...
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 ...
2
votes
2answers
920 views

Modified Cholesky factorization and retrieving the usual LT matrix

I have been looking at the modified Cholesky decomposition suggested by the following paper: Schnabel and Eskow, A Revised Modified Cholesky Factorization Algorithm, SIAM J. Optim. 9, pp. 1135-1148 ...
0
votes
1answer
87 views

One constrained optimization problem

Help me to find general solution of the following optimization problem: minimize $F(x,y)=(x^2-x_0)^2+y^2 $ subject to $y^2+x^2=Q$ where $x, y \in R$. Thanks in advice!
1
vote
0answers
97 views

Solving multiple phase angles for multiple equations

I have several equations and each have their own individual frequencies and amplitudes. I would like to sum the equations together and adjust the individual phases, ...
1
vote
2answers
361 views

How to calculate the Hessian of the Lagrangian at x and lambda

I'm working on a project that needs to solve a constraint optimization function. Currently, I'm using Knitro solver and it needs to calculate the the hessian of the lagrangian at x and lambda. I don't ...
3
votes
1answer
214 views

Optimization with constraint on solution of a linear system

I'm facing this optimization problem: $$\text{minimize} \quad a^T x$$ $$\text{s.t. the solution of $A(x) z + B(x) = 0$ belongs to a convex set $S$}$$ Here $A(x)$ is a linear matrix function of $x$ ...
1
vote
0answers
136 views

Nonlinear least squares and polygon area

I found this paper that describes preserving the global area of a polygon given some deformation (section 5): http://www.kunzhou.net/publications/2DShape.pdf I'm trying to do something very similar. ...
0
votes
1answer
129 views

invex functions - optimality functions

For a general convex program, a feasible point is an optimal solution if and only if it lies in a hyperplane whose a normal vector is the gradient to the objective function at this point. Please ...
2
votes
1answer
243 views

Finding a function that satisfies constraints numerically

I have the following system of equations for function $p(y)$ and I need help debugging my solution: $$\begin{align} 0&=\log(p(y))+1-\lambda-\gamma y^2-\eta ...
4
votes
1answer
528 views

Solving a set of 3 nonlinear equations with constraints

Problem statement: I am given 3 sets of equations that govern the force $P$, and also the neutral axis, defined by two variables, the radius from the center $r$ and also the rotation degree in ...
6
votes
2answers
194 views

How to solve mixed integer nonlinear programs?

I'm not a math expert so sorry for possible trivial questions. I have written this mixed integer nonlinear program (MINLP): $$ \begin{align} \min & \sum_{i \in ...
7
votes
3answers
416 views

Optimization problem for a parity-check code

I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
0
votes
4answers
430 views

Overdetermined set of quadratic equations

I am having a set of equations of the form $$\begin{align} (x - a_1)^2 + (y - b_1)^2 + (z - c_1)^2 &= W_1\\ (x - a_2)^2 + (y - b_2)^2 + (z - c_2)^2 &= W_2\\ & \vdots \\ (x - a_m)^2 + ...
3
votes
1answer
180 views

solving linear program with rank constraint?

I have a linear program where the variables are n vectors. Now I'd like to impose an extra constraint that k (k<=n) of the n vectors are linearly independent, or the matrix with the n vectors as ...
5
votes
2answers
2k views

Solving an overdetermined system of nonlinear equations

I'm wondering what the "best" way to approach solving a system of the following form would be: $A_1X + Be^{CY} = A_2$ $A_3X + Be^{CY} = A_4$ $A_5X + Be^{CY} = A_6$ etc. EDIT: Coefficients $A_i, ...
4
votes
2answers
213 views

discontinuous optimization

I'm solving the following problem: $$ \max_\rho \;\; \rho \; \min\left[\left( \frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$ where all constants and variables are defined ...
1
vote
1answer
51 views

Is there a name for this general problem (variation on least squares)?

The problem statement is as follows. Minimize $||g(X\beta)-y||^2$ with respect to $\beta$ where $g(\cdot)$ is some non-linear function, $y$ and $\beta$ are column vectors. General linear least ...
5
votes
1answer
325 views

How to optimize a rational function

Just a calculus problem: As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $$\begin{cases} 1 \leq a \leq c \\ 1 \leq b ...
1
vote
1answer
54 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
0
votes
1answer
177 views

Relaxation of a linear constraint in a quadratic programming problem

the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to ...
2
votes
1answer
212 views

Applied Math: Finding Roots

I am taking a Numerical Computation class, and we are currently learning about Newton's Method for finding the roots of a system of non-linear equations. I have no problems understanding how the ...