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1
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0answers
95 views

Solving multiple phase angles for multiple equations

I have several equations and each have their own individual frequencies and amplitudes. I would like to sum the equations together and adjust the individual phases, ...
1
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2answers
355 views

How to calculate the Hessian of the Lagrangian at x and lambda

I'm working on a project that needs to solve a constraint optimization function. Currently, I'm using Knitro solver and it needs to calculate the the hessian of the lagrangian at x and lambda. I don't ...
3
votes
1answer
208 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 $x$ ...
1
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0answers
135 views

Nonlinear least squares and polygon area

I found this paper that describes preserving the global area of a polygon given some deformation (section 5): http://www.kunzhou.net/publications/2DShape.pdf I'm trying to do something very similar. ...
0
votes
1answer
126 views

invex functions - optimality functions

For a general convex program, a feasible point is an optimal solution if and only if it lies in a hyperplane whose a normal vector is the gradient to the objective function at this point. Please ...
2
votes
1answer
239 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 ...
4
votes
1answer
506 views

Solving a set of 3 nonlinear equations with constraints

Problem statement: I am given 3 sets of equations that govern the force $P$, and also the neutral axis, defined by two variables, the radius from the center $r$ and also the rotation degree in ...
5
votes
2answers
181 views

How to solve mixed integer nonlinear programs?

I'm not a math expert so sorry for possible trivial questions. I have written this mixed integer nonlinear program (MINLP): $$ \begin{align} \min & \sum_{i \in ...
7
votes
3answers
412 views

Optimization problem for a parity-check code

I have $n$ data blocks and $k$ parity blocks distributed across $m$ boxes where each box can contain atmost $b$ blocks. Each parity block is Ex-or of some data blocks (for ease of understanding we can ...
0
votes
4answers
403 views

Overdetermined set of quadratic equations

I am having a set of equations of the form $$\begin{align} (x - a_1)^2 + (y - b_1)^2 + (z - c_1)^2 &= W_1\\ (x - a_2)^2 + (y - b_2)^2 + (z - c_2)^2 &= W_2\\ & \vdots \\ (x - a_m)^2 + ...
3
votes
1answer
176 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 ...
5
votes
2answers
2k views

Solving an overdetermined system of nonlinear equations

I'm wondering what the "best" way to approach solving a system of the following form would be: $A_1X + Be^{CY} = A_2$ $A_3X + Be^{CY} = A_4$ $A_5X + Be^{CY} = A_6$ etc. EDIT: Coefficients $A_i, ...
4
votes
2answers
212 views

discontinuous optimization

I'm solving the following problem: $$ \max_\rho \;\; \rho \; \min\left[\left( \frac{bn}{an-bm} \right)[(a-m)-\rho], \frac{b}{a}[a-(p+\rho)]\right]$$ where all constants and variables are defined ...
1
vote
1answer
51 views

Is there a name for this general problem (variation on least squares)?

The problem statement is as follows. Minimize $||g(X\beta)-y||^2$ with respect to $\beta$ where $g(\cdot)$ is some non-linear function, $y$ and $\beta$ are column vectors. General linear least ...
5
votes
1answer
322 views

How to optimize a rational function

Just a calculus problem: As a function of $K \geq 1$, what is the minimum value of $f/a + f/b + f/c + f/d + f/e$ subject to the following constraints? $$\begin{cases} 1 \leq a \leq c \\ 1 \leq b ...
1
vote
1answer
54 views

Transform the sample to make it more similar to a given

$X=\{x_{i}\}$ and $Y=\{y_{i}\}$ are numeric samples: $y_i \ge 0, x_i \ge 0, i \in [0..N]$. I need to find the mapping $F(X)=\{F(x_i)\}$ with fairly simple formula such that: Euclidean distance ...
0
votes
1answer
174 views

Relaxation of a linear constraint in a quadratic programming problem

the problem i have is like following: $x'Qx + f'x \rightarrow \min_x$ subject to $Ax \le 0$. $Q \ge 0$, so there's nothing wrong there, usual QP with a linear constraint. Is there a way to ...
1
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1answer
210 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 ...
0
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0answers
98 views

Linear Algebra Simplification Query

Apologies in advance; my linear algebra is not exactly up to scratch, but a program optimisation problem I've come across just feels like theres a better way mathematically rather than ...
0
votes
2answers
476 views

Using KKT to Write a Non-Linear Program

I would like to rewrite this two-level program as a Non Linear Program using KKT conditions $\min_{x \in X, \space y \in Y} \space f(x)$ $\text{s.t. } \space g(x, y) \leq 0$ $y \in \text{argmin}_{z ...
2
votes
1answer
293 views

Condition for existence of Lagrange-multiplier

Using the implicit function theorem one can prove the following: Let $X,Y$ be Banach-spaces, $U\subset X$ open, $f\colon U\to \mathbf{R}$, $g\colon U\to Y$ continuously differentiable function. If ...
2
votes
2answers
894 views

Using KKT conditions to maximize function

The goal is to maximize the following function: \begin{align} K_p(q) = q\log \frac{q}{p} + (1-q)\log \frac{1-q}{1-p} \end{align} where \begin{align} 0 \leq q \leq 1 \end{align} and $p \in (0,0.5)$ and ...
1
vote
1answer
110 views

Does a local minimum of a function always satify the Armijo rule

Does a local minimum of a function always satify the Armijo rule?
1
vote
1answer
183 views

Combinatorial Optimization Problem (can I/how do I solve this with integer programming?)

Inputs: 1) A set of M x N matrices, {A,B,C...N} containing only integers. 2) A single 1 x N matrix of floats, W (weights). I need to pull one row from each input matrix and sum values for each ...
1
vote
1answer
414 views

Optimizing Nonlinear Constraint Equations with Discrete Variables and Multiple Objective Functions

I have the following constraint functions: $$g_{i_{min}} \leq y_{i+1}-y_{i} \leq g_{i_{max}}$$ $$y_{i_{max}}-y_{i} \geq h_{i}$$ $$v_{i_{min}} \leq \Biggl[\frac{(y_{i+1}-y_{i})^{3} ...
0
votes
1answer
194 views

How to solve this problem programmatically?

Suppose we are given two real numbers: $a, b \in R$, $b > a$. Find $m \in [1, 2, 2.5, 5]$ and $k \in Z$, satisfying the following condition: $\frac{b-a}{m10^k} \in [5; 10]$. How to solve this problem ...
4
votes
2answers
416 views

Are the Karush-Kuhn-Tucker conditions applicable when one or more of the constraints are nonlinear?

I am just beginning to read about the use of "Concave Programming" methods and use of the Karush-Kuhn-Tucker conditions to identify the maximum value of a non-linear objective function subject to ...
1
vote
2answers
648 views

lagrange multiplier for more than 2 equality constraints

i couldn't do the following question for hours minimize $\sum_{i=1}^{n}x_{i}^{3}$ s.t. $\sum_{i=1}^{n}x_{i}=0$ and $\sum_{i=1}^{n}x_{i}^{2}=n$. by Lagrange multiplier rule ?
9
votes
2answers
418 views

A numerical optimization problem with a convolution in the constraint

I have a problem of the following form: minimize $\|Dx\|_2$ subject to $\|x*x\|_2 = 1$ where $x\in\mathbb R^n$, $D$ is a given diagonal matrix of positive entries, and $*$ represents convolution, ...
2
votes
4answers
527 views

Does Slater's condition hold for the following problem?

Does Slater condition hold trivially (because there are no inequality constraints) for the problem: $$\min_{x,y} \:\: cx+dy$$ s.t. $$e^x + e^y = 1.$$ Can I conclude there is a zero duality gap ...
1
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2answers
153 views

Find the range of $x$, given $y_{min} \leq y(x) \leq y_{max}$, where $y(x) $ can be any function ( Updated)

I have a series of inequalities: $$y_{1min} \leq y_{1}(x) \leq y_{1max}$$ $$y_{2min} \leq y_{2}(x) \leq y_{2max}$$ $$..$$ $$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$ Note that $x\in\mathbb{R}$ The ...
6
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7answers
2k views

Operations research book to start with

for somebody having a quite strong background in Mathematics, which are some good books for the domain of Operations research? I guess there are textbooks covering topics like linear and nonlinear ...
3
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2answers
427 views

A naive approach for nonlinear optimization on several real variables and one natural variable. Give examples of (potential) failure

Motivation: Example. To solve a problem on evaluating the maximum of a product of $n$ real variables subject to an equality constraint on its sum $S$ ($=100$), I used the Lagrange multipliers method ...