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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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Minimum-Time Cost Function

I'm trying to formulate an optimal control problem based off of this given Minimum-Time Cost Function: $$J(t_f)=\frac{1}{2}[x(t_f)-x_{des}(t_f)]^TP_f[x(t_f)-x_{des}(t_f)] + \frac{1}{2}\int_{t_0}^{t_f}(...
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Electron Groups and Platonic Solids

In my Chemistry class, lately I have been learning about Lewis Structures for molecules, and how the arrangements of groups of electrons on each molecule repel each other to form the molecule into a ...
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Production time optimization model

I have a problem where I have 2 different types of raw-material. These materials can be processed in two different production-lines which take different amount of time, depending on raw material. ...
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Can anyone help me maximizing profit

Hi guys i have a problem that i cant solve. its asking me to compute the optimum number of units that would maximize profit; if no more than 80 units can be built; and the resulting maximum profit. ...
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What is the meaning of stacked integers in linear algebra notation?

I was recently reading a stack exchange post on solving a polynomial regression problem with gradient descent, and one contributor identified the objective function as: $$\text{argmin}_{\beta} \| \...
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$L_{2,0}$ constraint in the optimization problem

I am trying to solve a minimization problem where my constraint is $$||W||_{2,0} = 1$$where $W \in R^{k \times k}$. The constraint is used in such a way that only one column of $W$ matrix is non-zero. ...
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Maxima of sum of two gaussians

I am trying to find an analytical form of the maxima of the function $$f(x) = a_1 e^{-b^2_1 x^2} + a_2 e^{-b^2_2 (x+x_c)^2} \ , \tag{1}$$ such that I can define a function $g(x)$ that has the ...
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Expectation Maximization (EM) : find all parameters from a PSF (Point Spread Function)

I have the 2 parameters arrays : $\theta=[a,b]$, $\nu=[r_0,c_0,\alpha,\beta]$ with the distribution (a point spread function = PSF = response of a Dirac) : $\text{PSF}(r,c) = \bigg(1 + \dfrac{r^2 + c^...
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MIQP problem slow to solve: how to rewrite it?

I am looking for suggestions on how to rewrite a MIQP problem. Let me firstly introduce the problem Notation: The unknown vector is $x$ with size $(4*2+225*2)\times 1$. We can think of the ...
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Hyperparameter optimization

How to choose hyper-parameters for optimisation methods in practice? I know that there are hyper-parameter optimisation techniques such as gradient-based or bayesian methods, but for instance it is ...
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Dynamic optimization - euler lagrange equations

I would like confirmation whether these assumptions are correct. I want to min: $\begin{equation} J=\sum_{i=0}^{N-1}\frac{1}{2}mg(2y_i+v_i) \end{equation}$ The dynamics are $ \begin{...
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Justification of the big M method

Given a LP we can solve it using the so called "big M method". I was curious about how this constant $M$ is determined and quickly enough ran into this. It is said that $M$ is large enough, so that ...
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Finding the minimum of function including a $-\frac{1}{x}$ term.

I would like to find the global minimum of the function: $$f(x, y) = 10(y^2-2x^3)^2 + (1-x)^2 - (y-1000)^{-1}.$$ Now my problem is the following. I know that $(1, \sqrt{2})$ is a local minimum, and I ...
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If a*b*c=8 then what is minimum value of (2+a)(2+b)(2+c) [on hold]

If $abc=8$ and $a,b,c >0$, then what is minimum possible value of $(2+a)(2+b)(2+c)$? Edit: I got the answer and have posted it below.
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What is the maximum value of $x^TAx$ subject to $x\in\{\pm1\}^n$?

Let $A \in \mathbb{R}^{n\times n}$ be symmetric and positive definite. What is the following maximum? $$\max_{x\in\{\pm1\}^n}x^T A x$$ My attempt: if all $a_{ij}\geq 0$, then \begin{equation} \...
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Chavtal-Gomory inequalities.

Let $X = \{x \in \{0,1\}^n | x_i + x_j \leq 1 \forall i \not = j\}$. Clearly, the inequalities $x_1 + \ldots + x_k \leq 1$ are all valid inequalities for $X$(for each $k \geq 3$). How can we obtain ...
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Maximizing a weighted sum of logarithms

I'm trying to show that for: $$p^* \equiv \max_{x} \sum^{k=n}_{k=1} a_k \ln x_k $$ Where: $ x \in \mathbb{R}^n \\ x \geq 0 \\ \sum_{k=1}^{k=n} x_k=c \\ c > 0 \\ \forall a_k, a_k > 0 \\ a \...
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Arguing that local minimum is global minimum

Let $$f(x)=\sum_{v=1}^k m_v\|x-x_v\|_2^2, \ \ \ x,x_v \in \mathbb{R}^n, m_v \in \mathbb{R}>0$$ Then $$\nabla f(x) = 2 \sum_{v=1}^k m_v(x-x_v) = 0$$ So $f$ has the local minimum $$x = \frac{\sum_{v=...
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degeneracy and duality in linear programming

I'm currently learning about linear programming and optimization methods and the most recent subject was duality I'm trying to understand the connection between degeneracy of the primal and ...
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Optimization Problem. Find Smallest Perimeter Given Area.

QUESTION Find the dimensions of a rectangle with area $1000$m$^2$ whose perimeter is as small as possible. MY WORK I think we are solving for $\frac{dy}{dx}$: \begin{align*} P &= (2x+2y) \\ ...
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Interview Question - Maximise Profit at Auction

You go to an auction to bid for a jar. You don't know exactly how much it is worth but it is uniformly distributed between 0 and 1000. You bid a price for the jar. If your bid is lower than the actual ...
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Algorithm to find integer combinations satisfying a set of inequalities

I have an engineering problem that is reduced to finding a set of positive integer combinations satisfying several inequalities and some other properties. Specially, let $\mathcal{S}$ be the set of ...
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Proving joint convexity in stochastic linear problems with fixed recourse

In Birge's stochastic optimization book, we have the following formulation where $\zeta$ is a random variable: $z(x,\zeta) = c^Tx + \min \{q^Ty|Wy = h -Tx, y\geq 0 \} \\ s.t. \ AX = b, x \geq0$ In ...
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“Completing the square” for Frobenius norm

Assume $k = <X-W,X-W>_F + <X-Z,X-Z>_F$ is it then possible to write $$argmin_X k = argmin_X norm(X-B)^2_F$$ where $B$ is some function of $W$ and $Z$? Note that F indicates Frobenius inner-...
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Knapsack Problem with Equal Weights

The problem consists in the standard knapsack problem in interger programming with the weights that all have the same values, for example they are all equal to one. It seems to me that the solution ...
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computing a subgradient regarding l1 matrix norm

The $L_1$ norm for matrices is defined by $||M|| = \sum_{i,j} |M_{i,j}|$. The question is to prove that the subgradient of $||M||$ is given by $sign(M) + A$ (the sign operation is applied on each ...
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Manipulation of Functions of Random Variables

Let $X:\Omega\rightarrow\mathbb{R}$ be a random variable such that $\mathbf{E}(X), \mathbf{var}(X)$ exist. a) Consider a function $f:\mathbb{R}\rightarrow\mathbb{R}$ defined: $$ f(t)=\mathbf{E}...
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Area Minimization of Circle

The radius of the circle having minimum area which touches the curve y=4-x² and the lines, y=|x| is? I tried using the normal to the curve and satisfying it with the centre of the circle but it just ...
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Help with matrix selection in matlab

I have this equalities, with $\Delta X$ and $\Delta S$ unknown matrices, $X$, $S$ and $\tau$ known. $$W^{-1}\Delta X W^{-1}+\Delta S=\tau X^{-1}-S$$or equivalently $$\Delta X + W\Delta S W=\tau S^{-1}...
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Minimizing a function with a given inner product

I'm trying to solve a question but a not sure on how to approach. Here is the question Let $V$ be a finite dimensional space with a given inner product and $A \subset V$ be a convex set and $a \...
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Eigenvalues in real and complex domain

I am modeling an optimization problem using semi-definite programming. The optimization variable is a rank-1 matrix $X=xx^T$. The vector $x$ contains the power network voltages, which are complex ...
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How do you set up this constraint in integer programming using binary variables?

Mike wants to invest in $X_1$ if and only if he invests into $X_2$ or $X_3$ or both. Please help i can't get my head around this Thanks
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Is there only one unique minimizer for this scalar complex quadratic form?

Consider $c,b \in \mathbb{C}$ and $f: \mathbb{C} \mapsto \mathbb{R}$, $$f(c) = bc' + b'c + cc'$$ is there only one extrema of $f$ corresponds to $$c^* = -b$$? where, $'$ means complex conjugate. ...
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Finding the nearest point to the origin on a line.

In the above picture, I am trying to find the smallest distance from points on the line segment to the origin. Now, I can see that it must be half the square root of 2, but this is not exactly my ...
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Prove that if the constraints to a linear program are integer, then the optimal solution is rational

I've got the following question that I can't quite figure out. I have a vague idea of how to do this. Attempt Assume, for contradiction, that the optimal solution $x*$ is not rational. This means ...
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Are we allowed to simplify the goal constraints in goal programming?

I know that we can't simplify the resource constraints while solving goal programming problems. But are we allowed to simplify the goal constraints?
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How to equally balance $\|X\|_1$ over the columns of $X$?

I have the following optimization $$\min_X f(X)+\lambda \|X\|_1,$$ in which $X\in \mathbb{R}^{n \times k}$, and $\|X\|_1=\sum_{i,j} |x_{ij}|$, and $f(X)$ is differentiable in $X$. It is ideal for me ...
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optimization exercise, linear algebra

Conjugate gradient method for minimalization: $f(x) = \frac{1}{2} x^{T}Qx − b^{T} x$ , where matrix $Q \in R^{n,n}$ is symetric and positive-definite. Method generated series $x_{k}$ in way: $$x_{k+1}=...
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A both strictly quasi-concave and quasi-convex function in $R^{2}_{+}$?

Is there an example of both strictly quasi-concave and strictly quasi-convex function in $R^{2}_{+}$? Thanks so much.
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Prettiest princess puzzle - probabilisticly finding maximum in a finite set with one scan

I've encountered a mathematical puzzle with an enlightening solution - sadly I recall only the outline of the solution without the specifics. The puzzle went somewhat as follows: A king with many ...
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Approximation of Inverse Hessian or Inverse Hessian Square Root times a vector

I know there are good methods for approximating a Hessian times a vector without actually forming the hessian. (Example here). Are there any methods of approximating the Inverse of hessian times a ...
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Maximizing perimeter to area ratio of a function

I was trying to find the curve, $y = f(x)$, from $x = x_i$ to $x = x_f$, constrained by $y(x_i) = 0$ and $y(x_f) = 0$, such that the ratio between the arc length of the curve and the area below the ...
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Can you prove this equality?

In their book An Introduction to Optimization, on the chapter on gradient algorithms, to prepare for discussing convergence properties of the descent methods, authors Chong and Zak have following: $...
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Linearization of two continuous variable [closed]

i have two continuous variable, that i want to linearized them. in the below equations v(t) and g(t) are the continuous variables, and hd(t) is a parameter. i want to implement this equations in the ...
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Can KKT be used in minimization s.t to constant param

Can KKT be used : min g(x) s.t x>=constant where constant > 0 I have read this The Kuhn-Tucker method: here says that This is an alternative, and ...
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Hessian vs. Bordered Hessian

I am confused as to whether to use the standard hessian or bordered hessian for the following problem. $f(x,y,z) = (x+1)^2+(y+1)^2+z^2 \text{ subject to } x+y=(x-y)^2 \text{ and } z-x-y=1$ We are ...
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$\partial U$ is a smooth curve (i.e., the image of a smooth path $\mathbf{c}$ with $\mathbf{c}' \not= 0$)?

In a section discussing global maxima/minima, my textbook says the following: Simply stated, $\mathbf{x}_0$ and $\mathbf{x}_1$ are points where $f$ assumes its largest and smallest values. As ...
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Solving multiobjective problem with matlb

I would like to solve a multiobjective problem with matlab with NSGA II procedure. The problem is a maximization/minimizationf objective functions. Can someone porvide me this code, and explain how to ...
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Question on Probability maximization

Mr A wants to join a Gamer's club. There are two identical boxes filled with Red and Green balls, and he has to pick up a green ball in order to join the club. You are required to allocate balls in ...
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Maximise $z = \frac{y}{2x+2y}+\frac{50-y}{200-2x-2y}$ given that $x+y$ is non zero and $x+y<100$. Also, $x\leq50$ and $y\leq50$ and non-negative.

Z is actually a probability function. I am finding where the probability is maximized. But I could find no way how to maximize this function. Original question is as follows: Mr A wants to join a ...