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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
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1answer
425 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
341 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 ...
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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 ...
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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
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0answers
173 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
599 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 ...
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2answers
268 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
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1answer
318 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 ...
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0answers
182 views
4
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0answers
294 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
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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, ...
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4answers
532 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
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1answer
487 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
465 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
187 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
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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
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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
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3answers
641 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 ...
-1
votes
0answers
108 views

Jacobian of Bilinear Cost Function

Could anyone help me with the Jacobian of: Given the following Matrices: ${E}^{1}, {A}^{21}, {A}^{22}, {C}^{1}, {A}^{12}$ $ F \left( {C}^{2}, {E}^{2} \right) = {\left \| {C}^{2}{E}^{1} - {A}^{21} ...
0
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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 ...
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0answers
716 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
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0answers
98 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
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2answers
446 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
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5answers
231 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
571 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
217 views

Optimization of multivariate non-linear function with linear constraint

I have this huge and ugly function. $$ f\left(x,y\right)= $$ $$ ...
1
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0answers
45 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 ...
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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
268 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
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0answers
302 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
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0answers
131 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
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0answers
232 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
389 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
133 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?
1
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1answer
716 views

Set of symmetric positive semidefinite matrices is a full dimensional convex cone.

If $S_n^+$ is the set of all symmetric positive semidefinite $n \times n$ matrices with entries in $\mathbb{R}$, how does it follow that it is a full dimensional closed convex cone in ...
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0answers
270 views

Can this non-linear optimisation problem be converted to a linear?

I have to minimize the function: $F(x)$ $F(x) = \sum_{i=1}^{M}||x_{i+1} - x_i - K(\frac{x_{i+1} + x_i}{2})||^2 + ||x_1-c_1||^2 + ||x_N-c_2||^2$ , where $x$ is a vector of $N$ scalars, $c$ are ...
1
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0answers
58 views

Nonlinear Optimization in Probability

Consider a locally-bounded function $f: X \times Z \rightarrow \mathbb{R}_{\geq 0}$, where $X \subset \mathbb{R}^n$ is compact and $Z \subseteq \mathbb{R}^m$ is closed. Let $m(\cdot)$ be a measure on ...
3
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3answers
267 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. ...
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0answers
140 views

Matlab: Solving $u_t = f(u) u'' + g(u) u' + h(u) u$

I am trying to solve this equation (numerically) $$\dfrac{\partial u}{\partial t}=\dfrac 3 x\dfrac \partial{\partial x}\left(\sqrt x \frac \partial {\partial x}(\nu(u,u^2,\cdots,u^k)u\sqrt ...
3
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2answers
154 views

Maximum of a product of a polynomial with positive coefficients and a finite sum of exponentials with negative coefficients on $[0,+\infty)$

Prove or disprove that $$ f(x)=\left(\sum_i a_i x^i\right)\left(\sum_j b_j e^{-\lambda_j x}\right) $$ where $\forall i, a_i>0$, $\forall j, b_j>0,\lambda_j>0$, and both sums are finite, ...
2
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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 ...
1
vote
1answer
144 views

Fixed point stability of piecewise linear system

I have an autonomous system of nonlinear equations of the form: $$Mx'' + C(\omega)x' + K(\omega)x + F_{nl}(x) = 0$$ where $M$ is the mass, $C$ the damping and $K$ the stiffness matrix. ...
1
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1answer
82 views

Least squares and (non-)linearity of parameters

I have a question about least squares and about what happens, if the function that we minimize, $E(P)$, is not linear in its parameters $P$. Assume we want to minimize a function (the exact terms are ...
0
votes
1answer
212 views

How to optimize entropy under linear constraints?

My problem is quite cumbersome. In general, it can be modelled as a non-linear programming problem, with linear constraints and non-linear objective function. The objective function is conditional ...