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 ...

learn more… | top users | synonyms

3
votes
1answer
33 views

existence of solution of $Ax= \max(b-x,0) $

How do you prove the existence of a solution to the linear system: \begin{equation} Ax= \max(b-x,0) \end{equation} A is an $n\times n$ matrix and $b$ is a vector in $\mathbb{R}^n$. $x$ is the ...
3
votes
2answers
168 views

Upper bound for Maximization problem

I have an optimization problem of the form Max $x_1+x_2+x_3+\cdots+x_n$ subject to $x_0^2+x_1^2+x_2^2+\cdots+x_n^2+x_{12}^2+x_{13}^2+x_{14}^2+ \cdots+x_{1n}^2+x_{23}^2 + \cdots +x_{2n}^2+ \cdots ...
3
votes
1answer
87 views

$P_{1c} = AP$ , $P_{2c} = BP$. How to find $P$? (being that $A$ and $B$ are $3\times 4$ matrices and $P$ is a $4\times 1$ vector)

This problem arose in my stereo vision project. $$ P_{1c} = A*P $$ $$ P_{2c} = B*P $$ where: $P_{1c}$ and $P_{2c}$ are $3\times1$ vectors, $A$ and $B$ are $3 \times 4$ matrices and $P$ is a ...
3
votes
1answer
277 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 $x$ ...
3
votes
2answers
29 views

Distance between a plane and set of points

Suppose $m$ data points belonging to a class represented by matrix $A$. Therefore, the size of matrix $A$ is $m\times n$. In addition, suppose $w\cdot x + b=0$ be equation of a plane in ...
3
votes
1answer
81 views

No clear analytic method to prove unique maximum? ($2^{-x}+2^{-1/x}$)

Prove that $f(x) = 2^{-x}+2^{-1/x}$ has the unique local maximum $(1,1)$ for $x>0$. Do not use computer software. Proving that $(1,1)$ is a maximum is easy, but I'm having trouble with the ...
3
votes
1answer
50 views

Non-linear system vs minimisation problem

If you have a non-linear system of equations which can be formally written as : \begin{equation} \begin{cases} F_1(\mathbf{x})=0\\ F_2(\mathbf{x})=0\\ \ \ \ \ \vdots\\ F_n(\mathbf{x})=0\\ \end{cases} ...
3
votes
1answer
190 views

fantasy basketball model

i'm creating a fantasy basketball model (could be used in other games too) where we can project how well a player will do against another team even when the player hasn't played against a certain team ...
3
votes
2answers
45 views

Sensitivity of polynomial global minimizers with respect to perturbations in the coefficients.

I'm trying to find the value of a global minimizers of a multivariate polynomial (4 variables) of high order numerically. The numerical values of the coefficients are coming from noisy measurements ...
3
votes
2answers
272 views

Maximizing “log det + log sum exp” function

I'm trying to find a numerical solution to the following optimization problem $$ \text{maximize } f(M) = \frac{1}{2} \log \det(M) + \log \sum_{i=1}^n \exp \left\{ - \frac{1}{2} x_i^T M x_i + a_i ...
3
votes
2answers
52 views

Max of $3$-Variable Function

I'm trying the find the maximum of the function $$f(a,b,c)=\frac{a+b+c-\sqrt{a^2+b^2+c^2}}{\sqrt{ab}+\sqrt{bc}+\sqrt{ca}}$$ for all nonnegative real numbers $a, b, c$ with $ab + bc + ca > 0$. I ...
3
votes
1answer
741 views

Lagrange Multipliers for Function Spaces

For some constant $A > 1$ I am trying to solve the constrained minimization problem minimize $F(u)$ in $C$ subject to $H(u) = 0$. Here $F(u) = \int -u dx$ and $H(u) = \int \sqrt{1 + (u')^2} dx - ...
3
votes
1answer
534 views

Lagrangian Multipliers

I have a fundamental question about Lagrange multipliers. Here it is: I have a function to maximize with respect to a parameter say $\theta$, subject to two constraints. Lets assume that the first ...
3
votes
3answers
289 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. ...
3
votes
1answer
283 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 ...
3
votes
1answer
258 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 ...
3
votes
1answer
52 views

Finding the root for a multivariate function.

Background In a practical problem I need to find the solution to: $$f(\bar{x}) - \bar{p} = \bar{0}$$ where $ f : \mathbb{R}^2 \rightarrow \mathbb{R}^2 $. I don't know the exact expression for $f$ ...
3
votes
0answers
52 views

Determining active constraints in KKT

Suppose there is a constrained optimization problem having inequality constraints. We can solve it using Karush-Kuhn-Tucker conditions. My question is how do we determine which constraints are active ...
3
votes
0answers
33 views

Solving an inverse squared sum

How would I go about solving this sum for $x$? $$\sum_i\frac{a_i}{(x+b_i)^2}=C$$ Where $\mathbf{a}$ and $\mathbf{b}$ are vectors and $C$ is a constant, and $x$ is a single number. It's for an ...
3
votes
0answers
42 views

Structural / design / meta optimization - is there mathematical theory. Optimization over categories?

There is huge branch of mathematical optimization theory, but it mostly considers the finding optimal parameter values for the predefined structures. There are variational calculus and optimal control ...
3
votes
0answers
166 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 ...
3
votes
0answers
39 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
73 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
822 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 ...
3
votes
1answer
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 ...
3
votes
1answer
48 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
130 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
26 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
90 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
83 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 ...
3
votes
0answers
284 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
240 views

Optimization problem with ratio objective

I need to solve the following optimization problem $$ \text{maximize} \quad \frac{(a^T x)^2}{x^TBx+c^T|x|} \quad \text{subject to} \quad \|x\|_1=1 \quad (\text{or alternatively} \quad c^T|x|=1), $$ ...
3
votes
0answers
795 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
4answers
86 views

Minimization on compact region

I need to solve the minimization problem $$\begin{matrix} \min & x^2 + 2y^2 + 3z^2 \\ subject\;to & x^2 + y^2 + z^2 =1\\ \; & x+y+z=0 \end{matrix}$$ I was trying to verify the first ...
2
votes
1answer
156 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
1answer
77 views

Question regarding KKT conditions in optimization

Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming. Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, ...
2
votes
2answers
1k 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
304 views

Sion's minimax theorem

Sion's minimax theorem is stated as: Let $X$ be a compact convex subset of a linear topological space and $Y$ a convex subset of a linear topological space. Let $f$ be a real-valued function on ...
2
votes
1answer
330 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
49 views

Why a convex cone cannot have more than one extreme point?

The way I define an extreme point is : A point which cannot be defined as a convex combination of two distinct points. I'm not able to extend this and show why a convex cone cannot have more than ...
2
votes
2answers
47 views

Solving for gradient of Frobenius norm term

Let's first define a couple of variables: $A,B,C \in \mathbb{R}^{m \times n}, D \in \mathbb{R}^{n \times n}$, and $\mu$ is a scalar. Say I have an ADMM sub-problem that looks like this: $\arg ...
2
votes
2answers
332 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
61 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 ...
2
votes
2answers
319 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
1answer
500 views

Definition of tangent cone in continuous optimization .

Looking at the definition of tangent cone in continuous optimization : If $M$ is a open subset of $\mathbb R^n$ $x \in M$, The tangent cone of $M$ at $x$ is defined by $$\mathbb T (M, x) = \big\{d ...
2
votes
2answers
64 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
702 views

Classifying local behavior of fixed points using eigenvalues from linear stability analysis of 3D system

I've learned about classification of fixed points of 2D systems using linear stability analysis and I am wondering how if at all I can apply the same process to analyzing local behavior of fixed ...
2
votes
1answer
71 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
590 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
74 views

Is standard eigenvalue optimization problem convex

For any arbitrary symmetric matrix A , is the standard eigenvalue problem convex $ \lambda_{max}(A)= \max_{\|x\| \leq1} x^{T}Ax$