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20 views

Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
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14 views

Maximizing the uniformity of density function subject to moment constraints

Background I want to find a probability measure for a continuous random variable, subject to moment constraints, that is maximally "uniform", as defined below: Definition: Maximally Uniform ...
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19 views

Constrained optimization using a cutting plane on a tetrahedron

Consider the figure below where $(a,b,c,d)$ is a tetrahedron and $p=(1-t)a+tb$ is a point on the $ab$ segment. If $n_a$ and $n_b$ are two unit vectors associated with $a$ and $b$, respectively, then ...
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2answers
46 views

How to solve this optimization problem? (may be gradient descent?)

I have the following optimization problem. $$\operatorname*{argmax}_{w} \|(1-w)\boldsymbol{X} -w\boldsymbol{Y}\|^2 \\ s.t. \quad 0<w<1 $$ How can I find the solution of this problem? May be ...
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0answers
49 views

Numerical nonconvex optimization problem

I have numerical data for the mapping $w:\mathcal{S}^{2+}\to\mathbb{R}$, where $\mathcal{S}^{2+}$ is $\{\mathbf{x}\in\mathcal{S}^2:x_3\ge0\}$, the 2-hemisphere on or above the $x_1-x_2$ plane. I ...
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1answer
30 views

Evolutionary algorithm

Can someone provide me a good reference for the CMA-ES algorithm? I'm new in the world of optimization and just reading the author reference doesn't help me a lot. I know the basic idea of a genetic ...
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2answers
157 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 ...
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1answer
19 views

squaring the equality constraints

When creating an unconstrained optimization problem from an equality constrained one, the usual way to build the Lagrangian, is by adding a term consisting of a multiplier, multiplied by the equality ...
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0answers
7 views

Get number equation using specific set of values for get given answer

I have do it for AI assignment. Need a logic for finding solution ..Here is the explanation of problem . I have answer ( any number like for example 10 ). And have some set of numbers (like for ...
1
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2answers
33 views

multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
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0answers
45 views

Software tools for medium-scale systems of polynomial equations

I am attempting to find all real solutions of a system of 12 polynomial equations in 12 unknowns. The equations each have total degree 6 and contain up to 1700 terms. I am only interested in real ...
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1answer
41 views

QR transformation with Householder transformation

It's a task i do to understand minimizing the error including the QR transformation with the help of Householder transformation. I think i really do something wrong but i dont get it running i hope ...
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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 ...
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0answers
25 views

Finding optimal hyperplane

I have a set of vectors $\{V_i\}$ in $n$-dimensional space. There is a number corresponded to each vector $\alpha_i = f(V_i)$ ($\alpha_i$ can be negative). I want to find a hyperplane which would ...
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0answers
12 views

Non-convex maxmin optimization

I am dealing with the following maxmin optimization problem: $c^*, x^* = \arg\max\limits_{c \in C, x \in X} [f(c, x) + \min\limits_{\tilde{x} \in X} g(c, \tilde{x})] $ $f$ and $g$ are differentiable ...
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0answers
34 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 ...
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0answers
67 views

Maximum Likelihood Estimation for State Space Models using BFGS

I have written some code that can do Kalman filtering (using a number of different Kalman-type filters [Information Filter et al.]) for Linear Gaussian State Space Analysis for an n-dimensional state ...
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0answers
19 views

Nature of Hessian of a function of a matrix

If input to a differentiable function is a matrix, what is the nature of Hessian of the function? Is it a tensor or something? This is a simple question, but I guess I am not sure where refer to, to ...
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2answers
128 views

Trace minimization of a matrix

Suppose $S = \pmatrix{1&1\\ 1&0\\ 0&1}$, $W$ is a $3\times3$ covariance matrix, which could be regarded as fixed. I need to find a $2\times 3$ matrix $Q$ that minimizes $$ ...
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0answers
30 views

Optimal numerial method for optimization of “Rosenbrock Banana”-like function

Which numerical methods would be optimal to find an extremum of a function with an almost flat "valley" (but a single minimum in the middle of the valley)? In this context optimal means the least ...
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2answers
55 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 ...
1
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1answer
32 views

Linear constraint in convex optimization

Is it true that the solution to a linearly constrained convex minimization problem can only be placed on the boundary of the constraint set, for any nonlinear convex objective, e.g. $$ \min_x f(x)$$ ...
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0answers
15 views

multivariate eigenvalue problem

Please, does anybody have an idea on how to solve a multivariate eigenvalue problem in a nonlinear system of equation. That is, we seek eigenvalues k of a matrix A where A is a matrix which contains ...
1
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1answer
98 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
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0answers
17 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 ...
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23 views

Writing down the KKT optimality conditions

Consider the problem Minimize $(1/2)\times{x}^{T}\times Q\times x+{P}^{T}\times x$ Subject to $(1/2)\times {x}^{T}\times P\times x+{d}^{T}\times x≤r$ Where Q and P are n×n matrices, P is ...
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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 ...
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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
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1answer
51 views

Linear Algebra quesion

$A^{-1} - \lambda A = B^{-1} - \lambda B - \alpha v v^T$ $A, B \in S^n_+$; $v \in R^n$; $\lambda, \alpha \in R_+$. Can we solve A in term of other variables?
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1answer
26 views

Estimating rates of convergence

If I have a set of data points obtained from a numerical approximation say 15.3828 15.2458 15.2095 15.2003 how can I estimate the rate of convergence?
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22 views

Find $\underset{\omega}{\min}$ $\underset{\beta \in \sigma(A)}{\max}$ $|\frac{\omega - \beta}{\omega + \beta}|$

As part of an algorithm for the solution of a linear system I'm trying to find $\omega > 0$, $\omega \in \mathbb{R}$ so that $$\underset{\beta \in \sigma(A)}{\max}|\frac{\omega - \beta}{\omega + ...
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0answers
25 views

Simultaneous iteration of Symmetric Matrices

Given a Matrix $A$ we can use Simultaneous iteration(Using power iteration on all columns simultaneously) to compute the d biggest eigenvalues. Now this method will give you the biggest eigenvalues, ...
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1answer
49 views

nonlinear least square

After some reading of Numerical Optimization, I am looking for some practice. Is there any practical resources of these problems. I mean with specific MATLAB code I've been looking but so far there ...
1
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0answers
39 views

Linear Programming, Optimal Solutions

I posted the whole question to give some context, but my problem lies with (iv). I think you're meant to use a formula for the generalization of the optimal solution, but I'm not really sure what ...
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1answer
39 views

Evaluate smartly a function on a multiplication grid

I am asking myself the following question: Suppose one has a grid $G \in \mathbb{N}^{n\times n}$ where $g_{ij} = i\cdot j$, $i,j \leq n$. I would like to evaluate a function $f: G \to \mathbb{N}$. ...
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1answer
37 views

Reference request: nonlinear systems, optimization, ode/pde

Could someone suggest me one or more good books on the following topics: Nonlinear systems: fixed point and Newton's method Optimization: steepest descent and Newton's-quasi newton methods ODE ...
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0answers
29 views

Hermite interpolation with interior points

I am trying to solve the following problem: Given the conditions on a curve c(u) of degree 4 at the points -1, 0, 1 as: c(-1) = 4; c'(-1) = 4; c(0) = 6; c(1) = -4; c'(1) = -6; find the generalized ...
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0answers
37 views

Quantitatively comparing event trains of different lengths for Poissonness

I have a parameterized, effectively black box process that generates a series of events (simulated action potentials). Different parameter values often lead to different numbers of events. How can I ...
1
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1answer
53 views

KKT point of a constrained optimization problem

Min$_{x}~x$ Subject to $x \geq 0$ For this problem, is $(x^{*}, \lambda^{*})=$$(0,0)$ a KKT point ? My try : I formulated corresponding Lagrangian and tried to find out the KKT point(s). ...
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1answer
103 views

calculate Jacobian matrix without closed form or analytical form

The question is probably clear in the title. In many of my applications mostly computer vision, I might not have the closed-form or analytical form of f (a multivariable function). It's calculated ...
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0answers
39 views

Maximize minimum optimization using linear integer programming

I am trying to solve a maximize minimum optimization. I have four different items that each of them has 10 values of Rates and for each value it has a corresponding weight. Then I have a free table ...
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0answers
17 views

Linear Quadratic Bilevel Programming Problem

How to solve this type of linear-quadratic bilevel programming problem ? Please help.
1
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0answers
65 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 ...
0
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0answers
17 views

Integration over a column vector

I have read in a Numerical optimization book (by Nocedal and Wright) that $f: \mathbb R^{n} \rightarrow \mathbb R$ is twice continuously differentiable and $p\in \mathbb R^{n} $ ...
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1answer
25 views

Lowest norm solution to a system of polynomial equations

I have a system of cubic equations: $$0=A_0+A_1 x+A_2 ( x \otimes x ) + A_3( x \otimes x \otimes x )$$ where $\dim A_0 = \dim x$ (so there are as many equations as unknowns). You may assume that the ...
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0answers
33 views

Linear least squares with sparse inequality constraints for support function estimation

The initial problem is the following: $$ ||h - h^{0}|| \to min \; \; s.t. Qh \leq 0 $$ where $h^{0} \in \mathbb{R}^{n}$ is known vector and $Q$ is a $m \times n$ matrix. The problem arises in specific ...
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1answer
34 views

How to know all errors in Optimization process?

I am using fminsearch in MATLAB to iteratively find the answer of a function. But it seems like the fminsearch cannot fulfill my expectation. So I ask this question whether there is any way to do what ...
0
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1answer
59 views

Can a 6-arm star be convex

Please help me with the following question. Suppose that the constant level contours of some function $V:\mathbb{R}^{2} \rightarrow \mathbb{R}$ have the shape of a symmetric 6-arm star. Can such a ...
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1answer
90 views

Does a Convex Function need to be Continuous

I have been trying the following problem and I am very confused. If possible the problem should be solved with derivatives. If the derivative exists for all the points on the graph then it is ...
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2answers
149 views

Quadratic Function must be positive definite to have a unique minimum

I have attempted the following question multiple times and I am very confused about the proof please help me solve it. Let $V(x)=a+b^{T}x+\frac{1}{2}x^{T}Cx$ for some $a \in \mathbb{R}, b \in ...