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Questions tagged [optimization]

Optimization is the process of choosing the "best" value among possible values. They are often formulated as questions on the minimization/maximization of functions, with or without constraints.

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How to minimize this energy by formulating it as a Poisson problem?

I have the energy function that I would like to minimize as: $$\sum_{\{i, j\}}((h_i - h_j) - q_{ij})^2$$ This is applied over a 2D grid, where $q_{ij}$ is the relative height between two cells $i$ $j$,...
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Minimize $E(m)=\sum_{i = 0}^{n-1} {m \choose i} \cdot [c \cdot (\frac{N}{m})^2]^i \cdot[1-c \cdot (\frac{N}{m})^2]^{m-i}$

Minimize over $m$ the expression: $E(m) = \sum_{i = 0}^{n - 1} {m \choose i} \cdot \left[ c \cdot \left(\frac{N}{m} \right)^2 \right]^i \cdot \left[1 - c \cdot \left(\frac{N}{m} \right)^2 \right]^{m ...
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How is this a convex constraint?

In one of the paper I found that for variable matrix $W$ where $W$ can be in the set of positive semidefinite matrices the following constraint is convex $$Tr(W)\geq c$$ where $c$ is some constant. In ...
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Formulation of SVM optimization problem

I need help in verifying/understanding a step in formulating an optimization problem used for support vector machines (though this question doesn't need any background in SVM). Consider a bunch of $m$ ...
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Convexity of Sum-of-max-linear-terms

Consider the sum-of-max-terms function $f: \mathbb{R}^m \mapsto \mathbb{R}$: $$\begin{align} f(z) = \sum_{k=1}^K \underset{j \in \mathcal{I_k}}{\max} \{z_j \} \end{align}$$ where $\mathcal{I}_{k} \...
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How to compute the derivative $f(X) = \|\mathcal{P}_\Omega(X-A)\|^2_F$?

How to compute the derivative $$f(X) = \| \mathcal{P}_\Omega(X-A)\|_F^2$$ here $\mathcal{P}_\Omega(\cdot)$ is a projector, $[\mathcal{P}_\Omega(Y)]_{ij} = Y_{ij}$ if $(i,j)\in \Omega$, zero otherwise....
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Deriving MLE of $\theta$ in $\text{Exp}(\theta,\theta)$ distribution

Suppose $X_1,X_2,\ldots,X_n$ are i.i.d variables having a two-parameter exponential distribution with common location and scale parameter $\theta$ : $$f_{\theta}(x)=\frac{1}{\theta}e^{-(x-\theta)/\...
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1answer
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Optimal Strategy in a money compounding model where one's interest is only consolidated at a fee

Say I have my main account with $ \$ 10000$ that gains interest at a rate of .1% a day. The interest collects in a separate account and I have to pay a certain fee, say $\$1$, to consolidate this ...
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Least squares solution to overdetermined AX=B where each matrix is a rotation

I can't seem to find anything that would help me with this particular problem. I have a bunch of measurements of a matrix A and corresponding matrix B, which I know are related by a third rotation X (...
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Is there any way to linearize $x-x^2\leq 0$?

I am trying to solve an optimization problem. The objective function and all constraints of this problem are linear except $x-x^2\leq 0$. Is there any way to linearize $x-x^2\leq 0$, where $x$ is a ...
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minimum amount of scrap [on hold]

Good day. I am creating an eCommerce site that sells lengths of beams by the mm, that needs to be cut from certain size beam that are held in stock. The aim is to have the minimum amount of off cuts/...
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LASSO's (or BPDN) parameter tuning

We have to solve the following problem : $$\min_x \|x\|_1 \text{ s.t. } \|Ax-y\| \le \sigma,$$ with $\sigma$ some positive real number, $A$ a complex rectangular matrix and $x,y$ complex vectors. ...
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Finding optimum time for change of equipment based on shift pattern available resources

I am trying to sort out a relatively simple problem, where I believe an algorithm or solving technique may already exist (Hungarian Algorithm?). I'd like to solve it using an methodology rather than ...
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What is the fault with Fritz John necessary condition for finding a local minimum for a general NLPP

The author says: It is also possible that, at some feasible point $x$, the FJ conditions are satisfied with Lagrange multiplier associated with the objective function $u_0 = 0$. In those ...
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Nonlinear optimization using fmincon: how to find optimal point for x vector

I am trying to solve maximization problem in which I have to find an optimal value for phi and phi is a not single-valued, it is a vector containing multiple values e.g. if n= 4 then I have to find 4 ...
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Point of tangency of a growing, quasi-convex function with constraint

I really tried to solve it but i couldn't. Also googled it but couldn't find it. Demonstrate that the point of tangency of a growing and quasi-convex function $$f \colon \mathbb{R}^{2} \mapsto \...
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Projection on to subspace with positive constraint?

I have an underdetermined system of 2 equations in n variables and a "starting point" $s$. I need to find the point satisfying the equations which is closest to the starting point. However, I have the ...
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What method will we use to find the optimal solution when we choose using the penalty function

I learn something about penalty method from these slides http://www.numerical.rl.ac.uk/people/nimg/course/lectures/parts/part5.2.pdf In the second slide,it show this quadratic penalty function $\Phi(...
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1answer
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Finding minimum of penalty-approximated quadratic problem

Find the minimum of the following quadratic function $$ f_{\alpha}(x) := \frac{1}{2} x^T H x + c^T x + \frac{\alpha}{2}(b^Tx)^2 $$ where matrix $H$ is symmetric and positive definite, and $\...
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glossary explanation of optimal method. Penalty method & Augmented Lagrangian method

In the optimization,i believe lots of people heard about Penalty method & Augmented Lagrangian method,but i wonder why are the creators use "penalty" and "Augmented" to name these method . ...
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1answer
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A question of primal-dual formulation

Suppose we have the primal problem: $\mathop{\min}\limits_{x\in X} F(Kx)+G(x)$, where $F,K,G$ are linear operators. And I want to ask how is the primal-dual formulation work to get the saddle-point ...
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Infimum of $\alpha\cos2\theta+\beta\sin2\theta+t_1\cos\theta+t_2\sin\theta$, if $\beta >0$ and $t_1^2+t_2^2=4$

For given $\alpha\in \mathbb{R}, \beta\in \mathbb{R^+}$, and $t_1^2+t_2^2=4$, we define the following function on $[0,2\pi]$: $$\varphi(\theta)=\alpha\cos2\theta+\beta\sin2\theta+t_1\cos\theta+t_2\...
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1answer
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KKT condition for the proximal algorithm

This slide shows that the KKT condition for the proximal gradient descent is this inequality. I don't know where this comes from. Using KKT , we can only get equality for the stationary condition, ...
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Dual of linear function with convex and non-convex constraints

I would like to compute the dual of the following problem by using the KKT conditions. However, due to form of the first constraint I am not able to obtain the dual. The problem is the following \...
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1answer
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help with Lagrange multiplier exercises [on hold]

Considering the following problems: Find the absolute extrema of $ f (x, y, z) = x + y + z \ $ in the set $$A = \{(x, y, z) \in R^3: x^2 + y^2 \leq z \leq 1 \}$$ Find the maximum and minimum ...
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Regression formula derivative

surely, this is an easy problem for a mathematician. Since I am just a student, I have some problems... I am trying to solve the following optimization problem: $$ \min_{w, \gamma} \frac{1}{2}\cdot(Aw+...
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How to form a perfect square?

You are given a square matrix $A \in \mathbb{ N}^{n,n}$, this number contains integers from $1$ to $n^2$. The task is to compute the minimal cost, to change this square into a perfect square. A ...
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controllable with airplane. [on hold]

Write the linear airplane equations as a four-dimensional linear system using the variables $x_l = a, x_2 = \phi, x_3 = \dot{\phi}$, and $x_4 = h$, and show that the system is controllable.
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Formulation of a constrained optimization problem with probabilities

we are given four probabilities for an event. We shall adapt these probabilities in order to maximize the entropy given a constraint $4 = \sum_{i = 1}^4 2p_ii \Leftrightarrow 2 = \sum_{i = 1}^4 p_ii$. ...
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Stochastic Optimization and Monte Carlo

Assume we are in a Brownian filtration where I denote $W$ the Brownian motion. My problem is to numerically compute $$ \min_X E (\int^1_0 X^2_tdt),\ \ \ \ (*) $$ where $X$ is adapted to the filtration ...
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A “divide-and-conquer” iterative procedure for minimizing a sum of convex functions

For simplicity, let's assume $f_i: \mathbb{R} \to \mathbb{R}$ is convex and define $$ g(x) := \sum_{i=1}^{n}f_i(x) $$ Suppose we want to compute $$ x^* := \arg\min_{x \in \mathbb{R}} g(x) $$ ...
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1answer
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Conic formulation with binary variables in Gurobi

I have a constraint of the following form $$x^2 \leq yz$$ where $z$ is binary, $y \geq 0$, and $x$ is free. Can Gurobi handle this constraint?
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Maximising linear function over a specific convex set of density matrices

All matrices being discussed in this question are density matrices, so they have the following properties: Hermitian Positive Semidifinite Trace = 1 We are currently in the space of all 4*4 density ...
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Optimization problem with variables in a sine function

I am trying to solve an optimization problem similar to this:- $$s: \min (\|H(s)g-g\|_2), $$ where $H(i,j)=\sin(k \pi (s_i - s_j))/\sin(\pi (s_i - s_j))$ and $g$ is my observation vector. Are there ...
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1answer
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Minimum of expression $\frac{16}{a} + \frac{1}{b}$

Let $a$ and $b$ positive real numbers such that $3(a^2 + b^2-1) =4(a+b)$. Find the minimum of expression $\frac{16}{a} + \frac{1}{b}$. I tried a geometric method but it doesn't work.
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how to constrain 2 matrix such that they don't cancel out in optimization

Given a simple generic optimization problem argmin y = (A+B)x + g(x) + epsilon Let's say we want to fix the matrix A, and allow for free-form optimization on B. What kind of regularization or ...
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0answers
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Minimum of a nested error propagation function

I'm a biology masters student working on a simple practical problem about some of the genetic tools we use. I'm very much a math novice, so please feel free to correct my notation/etc! I'm working ...
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1answer
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Local Maximum Point; Global Maximum Point

Given is the function: $f(x,y)=cos(x)+cos(y)$ Which of the following statements is correct? 1. The function has a local maximum point in $P (0, 0)$ This is correct, because the first order ...
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1answer
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Projected gradient descent

$\newcommand{\R}{\mathbb{R}}$There's the following problem that I found in a book: Let function $f:\R^2\to\R$ be defined as $$ f(x) = x_1 + x_1x_2 + (1 + x_2)^2 $$ Considering the feasible set $$ X ...
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How to check whether f is a convex and find all stationary points

Here is the problem What I attempted was calculating the Hessian and trying to prove that it's positive definite/semi-definite or negative definite/semi-definite. It doesn't seam to work as I am ...
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How many swaps in a set of size n will ensure that the set is shuffled reasonably well?

I'm implementing my own version of a shuffle method for shuffling a set of objects in a list. My implementation generates two (pseudo)random numbers and swaps the elements at these two indexes. ...
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Lagrange multipliers example with sympy - all minima but one maxima.

Consider the following optimization problem: Minimize $x^3+y^3$ Subject to: $x^2+y^2 \leq 1$ On the boundary of the constraint, we can consider $x=\cos\theta$ and $y=\sin\theta$. Then, the ...
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A question about the solutions of min-max problems

The minimax theorem states that for compact sets $X$, $Y$, if $f(x,y):X\times Y \rightarrow \mathbb{R}$ is convex for fixed $y$ and concave for fixed $x$, we have $$ \min_{x\in X}\max_{y\in Y} f(x,y) =...
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1answer
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Is Maxima/Minima of Lagrange function same as Maxima/Minima of function under consideration?

Given: To find critical points of function f(x) subject to constraints: g(x) = 0 We create a Lagrange function: L(x, λ) = f(x) - λg(x) Now, are the critical points and max/min of Lagrange function ...
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1answer
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Projection of functions in $L^1(\Omega)$ onto a $1$-dimensional subspace.

Suppose $f\in L^1(\Omega)$, where $\Omega\subset \Bbb R^n$ with $|\Omega|<\infty$. Let's consider a (probably not unique) constant function $c$ such that $$ \int_\Omega |f-c|\,d \mu = \inf_{t\in\...
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1answer
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Optimization: Absolut value function [closed]

I want to find the minimum of the following function, using a linear solver in Matlab: f = sum(((P * x - d)+|P * x - d|))*0.5*p) x (dimension [ix1]) is binary, P (dimension [nxi]),d and p are always ...
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1answer
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After reading the quadratic penalty method.i still don't understand what does it actually do,and the time of using it

After reading the quadratic penalty method.i still don't know what is this,take an simple question for example,this example is from page 491~492 of "Numerical Optimization" this book. http://www....
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1answer
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Find a permutation of the rows of a matrix that minimizes the sum of squared errors

I'm struggling with the following problem: Let $A, B \in \mathbb R^{n \times d}$. Denote by $\mathcal{P}$ the set of all possible permutations of the rows of $A$. Find a permutation $\pi \in \...
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1answer
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Change of variables in QCLP

Is there any change of variables that makes the following optimization problem easier to solve? \begin{align} \max_{x\in\mathbb{R}^n,t\in\mathbb{R}}\quad & c^\top x,\\ \mbox{s.t.}\quad\quad & ...
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Solve the optimization problem? [closed]

Minimize: $$F=x^2 + 2y^2$$ Subject to constraint: $x+3y=5$