The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
1answer
43 views

Concave optimal value?

Let $A \in \mathbb{R}^{n \times m}$ and $B \in \mathbb{R}^{m \times n}$. Consider a compact set $C \subset \mathbb{R}^n$. For all $x \in C$ define $$ f(x) := \min_{y \in \mathbb{R}^m} \{ x^\top A y ...
1
vote
1answer
155 views

Lagrange method - non-linear system of equations

I have to compute optimal parametres of truncated cone so that its Volume is fixed (lets say it is 1) and its surface is minimal using Lagrange method These are equations desribing my object: ...
0
votes
0answers
72 views

Max Quadratic Expression

Let $A \in \mathbb{R}^{n \times n}$, $A = A^\top$, $B \in \mathbb{R}^{m \times n}$, and $\mathcal{C} \subset \mathbb{R}^n$ be a compact, convex set. For $A$ not negative semidefinite, how to globally ...
1
vote
0answers
380 views

Significance of Rank of Jacobian

I have been struggling with this question. What is the significance of finding the rank of a Jacobian matrix of a function? I understand that the Rank of a matrix signifies the number of linearly ...
4
votes
2answers
526 views

Minimizing with Lagrange multipliers and Newton-Raphson

I am writing a program minimizing a real-valued non-linear function of around 90 real variables subject to around 30 non-linear constraints. I found handy explanation in CERN's Data Analysis ...
1
vote
1answer
517 views

A sufficient condition for a unique maximum of the product of two concave functions

Given two concave functions $f(x)$ and $g(x)$, what conditions in terms of these functions can ensure that $h(x)=f(x)g(x)$ have a unique maximizer on an interval $[a,b]$ for $a<b$?
0
votes
1answer
31 views

Second order least squares problem with one parameter

I want to fit a set of measured data $x_i$ and $y_i$ to the expression: $$ \beta^2 + \beta x_i = y_i $$ $\beta$ is my only free parameter. Although this is a really simple expression, the standard ...
2
votes
0answers
212 views

sufficient condition for KKT problems

For the Karush-Kuhn Tucker optimsation problem, Wikipedia notes that: "The necessary conditions are sufficient for optimality if the objective function f and the inequality constraints g_j are ...
0
votes
1answer
63 views

Finding the value which minimises all residuals

I have a series of observations, measurements made at various times $t$. I now need to determine the most likely value of $R$ (distance) using the model below. The guide says I should find the value ...
2
votes
1answer
142 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. ...
3
votes
1answer
402 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 ...
1
vote
1answer
258 views

Some type of Mixed Integer Nonlinear Programming Problem

This is a minimisation problem, to minimise the integral over possible $0\leq t \leq T$, $T$ is free, $$J = \text{min} \int_0^T (\alpha + \beta_1\cdot v \cdot R_T \cdot q+ \beta_2 \cdot ...
3
votes
1answer
432 views

Nonlinear Optimization/Programming: A good counter text

I am currently taking a nonlinear optimization course and the text is Bertsekas' "Nonlinear Programming 2e". I think the book does a decent job but I am a much more "hands-on" and visual learner so ...
2
votes
1answer
347 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 ...
1
vote
0answers
69 views

with a set of arbitrary initial values in newton method the system of equations don't have any answer

with a set of arbitrary initial values in newton method the system of equations don't have any answer. Does it mean that this system of equation don't have any answer at all or may be there exists a ...
1
vote
0answers
94 views

minimum of the function over symmetric body

Let $X$ be a normed space. Let $K$ be centrally symmetric convex body with $M$ known vertices and with $Vol(K)=V$. We subdivide body $K$ be $M$ equal pieces, $P_i, i=1, \ldots, M$, such that each ...
2
votes
0answers
174 views

Optimization over function spaces

There are many methods to solve optimization over standard spaces such as real and complex numbers or vector spaces of these. In my case I have a vector space of functions of a real value to find. ...
2
votes
3answers
600 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 ...
0
votes
2answers
274 views

Numerical optimization with nonlinear equality constraints

A problem that often comes up is minimizing a function $f(x_1,\ldots,x_n)$ under a constraint $g(x_1\ldots,x_n)=0$. In general this problem is very hard. When $f$ is convex and $g$ is affine, there ...
1
vote
1answer
319 views

Comparison of nonlinear system solvers?

I am dealing with nonlinear systems of equations that I am trying to solve numerically. These sets of equations derive from structural mechanics involving strong nonlinearities, like contact. The size ...
0
votes
0answers
182 views
4
votes
0answers
295 views

(easier version)What is a working example for finding the maximum of a algebraic function $f(a,b,c,d,e)$ with 4 equality …

edit 2: If you are founding an answer would be interesting to you, please upvote this question, because i may use those point in order to start a new bounty. Edit 3: Can somebody answer? Edit 4:I ...
1
vote
0answers
20 views

Finding an element in a very specific set

I ran into the following problem during some self-motivated studies, and for the last 24 hours I have been unable to solve this problem. The problem arose by itself, meaning it doesn't have a source, ...
4
votes
4answers
536 views

Summary of Optimization Methods.

Context: So in a lot of my self-studies, I come across ways to solve problems that involve optimization of some objective function. (I am coming from signal processing background). Anyway, I seem to ...
0
votes
1answer
490 views

Max function is continuous and concave-convex?

Let $A \subset \mathbb{R}^n$ and $M \subset \mathbb{R}^{n \times m}$ be discrete sets of cardinality $N \geq 1$. Consider the function $f: \mathbb{R}^n \times \mathbb{R}^m \rightarrow \mathbb{R}$ ...
2
votes
1answer
470 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 ...
0
votes
1answer
188 views

Maximize function of two variables

Let $f: X \times Y \rightarrow \mathbb{R}$ be a continuous function, where $X \subset \mathbb{R}^n$ and $Y \subset \mathbb{R}^m$ are compact sets. Say under which conditions we have that $$ \max_{x ...
0
votes
0answers
126 views

MiniMax Theorem

Consider the compact sets $X \in \mathbb{R}^n$, $Y \in \mathbb{R}^m$, $A \in \mathbb{R}^n$, $M \in \mathbb{R}^{n \times m}$. For fixed $(\bar{a},\bar{B}) \in A \times M$, by the MiniMax Theorem we ...
0
votes
1answer
99 views

Solving an optimization problem involving reciprocals

I am trying to solve the following minimization problem, perhaps by getting it into a LP form: Let $u= [u_1, u_2, ...u_N]^T$ a column vector, and $v=[{1\over u_1}, {1 \over u_2}, ...{1 \over u_N}]^T$ ...
0
votes
2answers
44 views

Solving a grid full of variables from the totals

We've been stumped by a much larger version of this problem, but we have simplified it down to a simple example: ...
2
votes
3answers
648 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 ...
0
votes
0answers
84 views

Optimization with Tricky First derivative condition

I have a function $f$ such that the first derivative consists of $f'_i=\alpha_i*\beta_i$, where if $\beta_i=0$ then I get linear dependence in my solution (which is not allowed). So for the first ...
0
votes
0answers
39 views

A Nonlinear Optimization Problem

Given $a_1,a_2,a_3,b_1,b_2,b_3,c_1,c_2,c_3,e$ are real, $v_1,v_2,v_3$ are unknown, and $$a_1(v_1-b_1)^2+a_2(v_2-b_2)^2+a_3(v_3-b_3)^2=e,$$ find the smallest value of ...
1
vote
0answers
718 views

Simple example application of Karush-Kuhn-Tucker conditions to minimization problem

I am wondering if there is a simple example application of the Karush-Kuhn-Tucker conditions to show that a minimum exists for a multivariate minimization/optimization problem. Could anyone suggest a ...
0
votes
0answers
99 views

Iterative scheme for a nonlinear optimization problem

Let $\mathbb{PD}_3 \subset \mathbb{R}^{3 \times 3}$ be the set of the positive-definite $3 \times 3$ real matrices. For given $v \in \mathbb{R}^{3 \times 1}$, consider the function $f_v: ...
1
vote
2answers
447 views

LP relaxation for ILP\IP (integer linear programming)

I am familiar with LP relaxation for ILP (or IP). Assume we concern with integer minimization problem, which we formalize using ILP; we then relax the ILP into LP and we say that the LP provides a ...
1
vote
5answers
232 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
1answer
573 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
218 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
271 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
304 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
votes
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
234 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
394 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
136 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?