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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
1answer
36 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 ...
3
votes
1answer
45 views

How to find the minimal value of this function under such constraint

$f(x,y)=x-\sqrt{y-x^2}$ with $a<x<b$ and $x^2<y<c*x-d$. What I did is, first take partial derivative at $x$ and $y$ respectively, however, there is no critical point because fy is always ...
3
votes
1answer
71 views

Does having a zero eigenvalue preclude a matrix from being indefinite?

If a $3\times3$ matrix has a positive eigenvalue, a negative eigenvalue, and a zero eigenvalue, is it then, by definition, indefinite? I think so, since the matrix has both a positive and a negative ...
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
3answers
93 views

How do one solve a nonlinear combinatoric problem?

I am an undergraduate CS student and I am struggling with a problem. $Qx = b$ where $Q$ is a constant $m \times n$ matrix (with $m>n$), $x$ is a $n \times 1$ vector and $b$ is a $m\times 1$ ...
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 ...
3
votes
1answer
224 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
2
votes
1answer
146 views

Prove or disprove the conjecture about the function below.

After thousands of numerical tests we stated the conjecture that their is exactly one local extremum of the function below. $$ {\rm f}\left(w\right) = {1 \over 2}\sum_{i = 1}^{n}\left({1 \over 1 + ...
2
votes
3answers
672 views

Constrained optimization: equality constraint

I have this very general problem (for $n>2$): $$ \begin{align} & \max Z = f(x_1,\ldots ,x_n) \\[10pt] \text{s.t. } & \sum_{i=1}^{n} x_i = B \\[10pt] & x_i \geq 0 \end{align} $$ Assume ...
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 ...
2
votes
2answers
81 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. ...
2
votes
2answers
146 views

What change of variables, if any, transforms this nonconvex problem into a convex one?

I'm looking for a convex reformulation, if any exists, of the following minimisation problem: Let $A$ be a symmetric, positive definite $n \times n$ matrix, and $b \in \mathbb{R}^n$. Minimise ...
2
votes
2answers
58 views

$\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$

Solve $\sum\limits_{k=0}^{19} \sqrt{1+u_k^2} \rightarrow \min$, such that $x_0 = 0, x_{20} = 5$ and $x_{k+1} - x_k = u_k$. I think I know how to solve problems like these recursively, but I ...
2
votes
2answers
439 views

How can I solve Lagrange multiplier equation with multi constraints?

This site is really awesome. :) I hope that we can share our ideas through this site! I have an equation as below, $$ min \ \ w^HRw \ \ subject \ \ to \ \ w^HR_aw=J_a, \ w^HR_bw=J_b$$ If there is ...
2
votes
1answer
69 views

Maximalization of a cubic puzzle

What is the maximal volume of a post package of length $L$, width $W$ and height $H$, subject to the following restrictions: $L+W+H \leq 90 $ $L \leq 60$, $W \leq 60$, $H \leq 60$ Intuitively I ...
2
votes
1answer
355 views

Nonlinear optimization with rotation matrix constraint

I'm trying to optimize the equation || R - W || = minimum where W is a predetermined 3x3 matrix and R is the 3x3 matrix that I'm trying to optimize, with the ...
2
votes
1answer
227 views

Optimizing trigonometric and nonlinear functions

First, Please, keep in mind that I'm a programmer not mathematician, and I have a fair mathematical background. I used optimization in Java to fit some observations to a trigonometric function, I ...
2
votes
1answer
155 views

non linear optimization

How to solve this optimization problem using matlab or some other tool. I know that, this is a convex problem with non-linear constraint $\rho\geq \rho_{min}$ , so i have tried many times it in ...
2
votes
1answer
556 views

Prove every local minimum is a global minimum

Let $Q\in\mathbb{R^{dxd}}$ and $A\in\mathbb{R^{d'xd}}$ be two matrixes and $b\in\mathbb{R^d}$, $c\in\mathbb{R^{d'}}$. Suppose $d'\lt d $. For $x\in\mathbb{R^d}$. Minimize $$f(x)= ...
2
votes
1answer
995 views

Prove the supremum of the set of affine functions is convex

Let $\langle f_i \rangle _{i \in I}$ be a family of affine functions on a convex and compact set $\Omega \subset \mathbb{R^d}$ such that $f_i = a_i.x +b_i$ for $x \in \Omega$. Prove that f, defined by ...
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}& ...
2
votes
2answers
922 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 ...
2
votes
1answer
298 views

Condition for existence of Lagrange-multiplier

Using the implicit function theorem one can prove the following: Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If ...
2
votes
1answer
50 views

Eliminate 2 variables from 3 equations with lots of parameters

I want to eliminate the variables x and y from these 3 equations in a way that all parameters appear in one equation without x and y: ...
2
votes
2answers
591 views

Which optimization algorithm converges faster?

everyone. I'm having a large scale unconstrained optimization problem. If I treat the unconstrained problem as a constrained problem with infinity constraints, I should be able to use both the ...
2
votes
1answer
68 views

Formulate the Langrangian function of a non-linear optimization problem and solve it for $y\geq0$

Consider the (non-linear) optimization problem ($P$) $$max \quad3x_1 + 4x_2$$ $$s.t. \quad x_1^2 + x_2^2 \leq 25$$ $$ \quad x_1,x_2 \geq 0$$ Formulate the Lagrangian function ...
2
votes
1answer
628 views

Python numerical solution for a nonlinear second order ODE with two boundary conditions

I want to solve numerical the next equation, in Python $$u''(x) = \left( a - \Big(b\big(u(x)^{2}\big)\Big) \right) \big(u'(x)\big)^{3}$$ it is a nonlinear second order $ODE$ with two $B.C$. ...
2
votes
2answers
133 views

optimality of quadratic programming problems

Suppose we have a general quadratic programming problem: \begin{align} \min_{x}\,\,&c^Tx+\frac{1}{2}x^TQx,\\ \mbox{s.t.}\,\,& Ax=b,\\ &x\geq0, \end{align} where $Q$ is positive ...
2
votes
1answer
121 views

Control on Conformal map

Let $\Omega$ be smooth simply connected open set of $\mathbb{R}^2$ such that $\overline{\Omega}$ is compact. We know that there exists a conformal diffeomorphism $\psi$ from $\mathbb{D}$ to $\Omega$. ...
2
votes
1answer
105 views

Nonlinear optimization question

For (x,y) in $\mathbb R^2$, consider f(x,y) = $x^2 -2xy + \frac{4}{3}y^2 - 4y$ Find the local minimum of f. Is it a strict local minimum? Compute the $\lim\limits_{|(x,y)|\to \infty}$ f(x,y) to decide ...
2
votes
1answer
143 views

Positive values for a set of quadratic forms of Hermitian Matrices. (To find a set of vectors in which a hermitian matrix is positive definite)

Assume all matrices I discuss about are $N \times N$ and the vectors conform with dimensions. Consider the following set of Quadratic inequalities where all the matrices $A_i$ are hermitian. ...
2
votes
3answers
610 views

how to compute the gradient of a function at an extremal point

I am writing a computer program that searches for the minimum of a multivariate function $f: \mathbb{R}^n \to \mathbb{R}$. This function is in fact the sum of many functions: $$f(x) = \sum_{i=1}^m ...
2
votes
1answer
480 views

When $\min \max = \max \min$?

Let $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ be compact sets. Consider a continuous function $f : X \times Y \rightarrow \mathbb{R}$. Say under which condition we have $$ \min_{x \in ...
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 ...
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 ...
2
votes
2answers
902 views

Using KKT conditions to maximize function

The goal is to maximize the following function: \begin{align} K_p(q) = q\log \frac{q}{p} + (1-q)\log \frac{1-q}{1-p} \end{align} where \begin{align} 0 \leq q \leq 1 \end{align} and $p \in (0,0.5)$ and ...
2
votes
4answers
530 views

Does Slater's condition hold for the following problem?

Does Slater condition hold trivially (because there are no inequality constraints) for the problem: $$\min_{x,y} \:\: cx+dy$$ s.t. $$e^x + e^y = 1.$$ Can I conclude there is a zero duality gap ...
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
1answer
83 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 ...
2
votes
1answer
26 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. ...
2
votes
1answer
49 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: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
2
votes
1answer
46 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
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
1answer
85 views

Extremum of functional of a complex function

consider functional $E$ defined by $$E[z]=\int F(x,z(x))dx$$ where $F$ is a complex-valued nonlinear function. How can we find the function $z(x)$ so that $$G=|E|^2=EE^*=\iint ...
2
votes
1answer
34 views

Conceptual Understanding of Non-Linear Optimization Problem

I'm in non-linear optimization, and I'm having trouble wrapping my head around what this problem is asking me for. If anyone could help with a conceptual explanation (not an answer!), it'd be greatly ...
2
votes
1answer
54 views

Initialization of Limited-memory BFGS (using libLBFGS)

I am using the package libLBFGS in order to minimize an objective function, for which the first derivative (with respect to the optimization variable) is known and computable. I use the default ...