For questions about optimization and optimization problems with non-linear restraints.

learn more… | top users | synonyms

0
votes
2answers
56 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 ...
0
votes
0answers
38 views

Finding a solution using the principle of maximum entropy?

I have set of linear constraints and would like to find an answer to its unknown variables, $p_i$'s. One of my options to find a solution for $p_i$'s using maximum entropy problem, $\max(\sum - p_i ...
0
votes
0answers
106 views

Maximize the maximum Eigenvalue under a diagonally constrained matrix

Suppose we have $N\times N$ Hermitian matrix $\mathbf{A}$ I want to find the real $N\times N$ diagonal matrix $\mathbf{D}$ that maximizes the sum of the maximum Eigenvalues : $\mathbf{D}=\arg\max ...
2
votes
2answers
94 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. ...
0
votes
0answers
25 views

min max of a rational function

The variable is a vector $x \in \mathbb{R}^n \times\mathbb{R}^m_+ \times \mathcal{E}$ where $\mathcal{E}$ is an ellipsoid of dimension $e$. I would like to find the min and max of the following ...
2
votes
1answer
56 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: $$f(\mathbf{w}\mid\mathbf{p},\mathbf{q})=\sum_{t=0}^T \left[\left(\sum_{i=0}^I w_ip_{ti}\right)-q_t\right]^2,$$ ...
3
votes
1answer
72 views

Are all non-convex problems created equal?

The distinction between convex and non-convex problems is usually dubbed as the distinction between easy and hard problems. While in the convex case you are golden (local optima are global optima; ...
1
vote
1answer
45 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 ...
0
votes
1answer
27 views

Can $\min f'x$ s.t. $(a'x - b)^2 \le d $ be written as a SOCP?

It does not appear to be significantly different from the form listed here: http://en.wikipedia.org/wiki/Second-order_cone_programming with (in article notation) $i = {1}$, $ A = a$, and $b$, $d$ as ...
0
votes
0answers
27 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 ...
2
votes
1answer
58 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
59 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 ...
0
votes
1answer
51 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 ...
0
votes
0answers
21 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
votes
1answer
38 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 ...
0
votes
1answer
32 views

Implement $\max$ with a closed form expression?

I have 2 functions: $f(x)$ and $g(x)$. Both of them range in $[0,1]$. Is there some way to define a $h(x)$ that efficiently takes the greater one of $h(x)$ and $g(x)$, i.e. $h(x) = \max \{ f(x),g(x) ...
1
vote
1answer
63 views

Showing a function is concave

Given $F(\underline{x}) = Ax_1 + Bx_2 + \ln(a^2-(x_1^2+x^2_2))$ on $S=\{\underline{x}\in\mathbb{R}\mid x_1^2+x_2^2<a^2\}$ with $A,B,a\in\mathbb{R}$, show that $F$ is concave on $S$. Since we have ...
1
vote
0answers
39 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
44 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 & ...
0
votes
1answer
70 views

Lagrange condition and second-order conditions

Given a function to minimize or maximize with equality and/or inequality constraints, I can use Lagrange multiplier and/or KKT to solve such problems. So I understand how it works. My problem is ...
0
votes
0answers
51 views

Linearization of multiple normal functions

I have noticed that it takes a very long time to perform non-linear least squares fitting on datasets similar to this: where there are multiple Gaussian distributions to be fit to experimental ...
4
votes
1answer
45 views

QR-Decomposition of matrix valued function

I already posted the following question on MO, but id did not raise much interest there. Maybe the title is too elementary to gain research interest. Suppose I have a matrix valued function $$ ...
1
vote
0answers
15 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?
0
votes
1answer
47 views

Gradient of Objective Function

I want to know how to calculate the gradient $\triangledown f\left ( \mathbf{x} \right )$ of this functions: $f\left ( \mathbf{x} \right )=\left | \mathbf{a}^{H}\mathbf{x} \right |^{2}$, ...
1
vote
2answers
71 views

Optimization of Product of Different Objective functions (Ex.: Maximize The Product of projections of a complex vector)

Suppose We have this optimization problem which is convex $\mathbf{x}={\arg}\: \underset{\mathbf{x}}\max f_{i}\left (\mathbf{x} \right )$ But the product of different objective function is ...
0
votes
0answers
30 views

minimizing sum of squares

Say i have the following optimization problem. min $\sum\limits_{i=1}^m \parallel r - (y_i - Rx_i) \parallel_2^2$. where we are optimizing over $r \in R^n$ and also $R \in R^{n, n}$ is given. Also, ...
1
vote
0answers
29 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?): $2x=\lambda _12x$, ...
5
votes
3answers
114 views

Find min of $\frac{\left( \sum kx_k \right) \left( \sum x^2_k \right)}{\left( \sum x_k \right)^3}$

With $n \ge 2$ and $x_1,\ x_2,\ \dots,\ x_n > 0$. Find the minimum of: $$ M = \frac{(x_1 + 2 x_2 + ...+ nx_n)( x^2_1 + x^2_2 +...+x^2_n)} {\left( x_1 + x_2 +...+ x_n \right)^3}$$ For specific ...
1
vote
3answers
149 views

Minimizing sum of squared distances from point to spheres

Given some spheres with known radius and known origin in three dimensional space, I want to find the point P that lies "closest" to all these spheres. The meassure of closeness, I guess, will be the ...
0
votes
1answer
15 views

Sets of feasible directions

I'm not exactly sure how the different points matter. I believe $p=[1,1,-1]^T, [2,-1,0]^T, [3,0,-1]^T, [0,3,-2]^T$ are all feasible directions.
0
votes
1answer
48 views

Maximizing the product of projections of a vector on another vectors

I want to get the $N\times1$ complex vector $\mathbf{x}$ which maximizes this real valued function $f=\mathbf{x}^{H}\left (\mathbf{a}_{1} \mathbf{a}_{1}^{H}\mathbf{x}\mathbf{x}^{H}\mathbf{a}_{2} ...
1
vote
1answer
36 views

Impact of constraints on convex optimization

Let me start by saying I know almost nothing about optimization so please bear with me. Basically, I am wondering whether it is possible to solve a problem with two constraints by solving the problem ...
0
votes
0answers
29 views

Hessian of a non-linear Matrix function

Apologies if this is a silly question, but I am really confused. I am trying to find the Hessian of a non-linear function $f$. I understand that the Hessian of $f$ with respect to $A$ is the Jacobian ...
1
vote
0answers
15 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 ...
0
votes
1answer
32 views

Finding the maximum/minimum of a homogeneous function on $R^n$

Suppose that $f:R^n\to R$ is homogeneous. Also, suppose that the $argmin_xf(x)$ is non-empty. Is it true that if there exist $x^*\in R^n$ such that $f(x^*)=0$, then $x^*=argmin_xf(x)$?
1
vote
2answers
30 views

Nonlinear optimization in exponents

$$ max \pi = 4x_1^\frac{1}4x_2^\frac{1}3 - x_1 - x_2 $$ It is not difficult to determine that this function is concave and yields a global maximum at some point for the quantities $ x_1, x_2 >= 0 ...
1
vote
1answer
65 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). ...
0
votes
1answer
41 views

Convex functions -> quasi-convex functions -> … can we weaken the assumptions?

First of all let me say that I'm new to optimization. I realized that quasi-convex functions share with convex functions some nice properties, so I wonder if we can push the weakening a little ...
0
votes
0answers
37 views

Unconstrained Nonlinear Nonconvex Optimization: LBFGS vs. Interior Point Methods?

I'm finding the literature on interior point methods somewhat inaccessible but I've found papers benchmarking different interior point methods for unconstrained nonlinear Nonconvex optimization. I ...
1
vote
1answer
43 views

least-square optimization with linearly depend solution $x$

What is the exact solution $x_{n \times 1}$ of the following constrained optimization problem \begin{align*} &\min \|A x - b\|^2 \\ s.t.& C x = 0 \end{align*} where $A$ is full column rank $m ...
1
vote
2answers
60 views

Minimizing a convex cost function

I'm reviewing basic techniques in optimization and I'm stuck on the following. We aim to minimize the cost function $$f(x_1,x_2) = \frac{1}{2n} \sum_{k=1}^n \left(\cos\left(\frac{\pi k}{n}\right) x_1 ...
0
votes
1answer
71 views

Can SVD help to solve (inequality) constrained least squares problem?

Consider the following minimization problem: $$ ||Q u - h^{o} ||^{2} \to min \;\;\; s.t. \; u \geq 0 $$ where $Q$ is $m \times n$ matrix and $u$ is $n$-dimensional vector and $h^{0}$ is ...
0
votes
1answer
23 views

Non-linear estimate parameter

I have one non-linear function that define $$E_x(a,b)=\int K_\sigma(y-x) \cdot(b-b. e^{-a\cdot f(y)} \,) dy$$ where $y$ is neighboor points of $x$; $f(y)$ is a function of $y$; and $a$ is constant. ...
0
votes
0answers
19 views

Linear Quadratic Bilevel Programming Problem

How to solve this type of linear-quadratic bilevel programming problem ? Please help.
0
votes
1answer
31 views

Noncontinuous subadditive function

Is there any non-continuous additive function $f(x+y)= f(x)+f(y)$ from $\mathbb R$ to $\mathbb R$?
0
votes
0answers
55 views

Expressing rank condition of a matrix in terms of its elements

Let $x \in \mathbb{R}^{n}$, define $X = xx^{T}$. I have an optimization problem with some linear constraints and few quadratic constraints, and I have to solve for $x$. Using $X$ as the unknown ...
1
vote
0answers
70 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 ...
2
votes
0answers
89 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 ...