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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Finding the Point at Which a Function is a Maximum when the Derivative is a Transcendental Function

Take $$ f\left(x\right)=-x\left( x\sqrt{4-x^2}-4\arccos\left(\frac{x}{2}\right) \right)\arccos\left(\frac{x^2+d^2-1}{2dx}\right) $$ and try to find the point where $f$ is at a maximum, given $1< ...
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Find the point on the parabola $2y=x^{2}$ that is closest to the point $(-4,1)$

The first part of the derivative which is to the power of $-1/2$ is too small to be considered relevant. I'm not sure how to proceed from here. The answer is $(-2,2)$ but I am not sure how to get ...
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Sorting for maximum mean squared successive difference

I have a set of numbers and I have to order them for maximum MSSD (mean squared successive difference). For example, if I have the ordered set {1,2,3,4,5,6} this would give me an MSSD of ...
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Minimizing a function with absolute value in one variable

Minimize the following scalar function: $f(x)={\frac 1 2}(x-b)^2+\lambda|x|$ where $b$ and $\lambda \ge 0$ are given real parameters. I need to show that the function is convex and find the unique ...
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Find lone peak with high sampling cost

(I'm not sure if this is the correct stackexchange place to ask; please redirect me if it doesn't belong here.) I have a 2D function f(x,y) which is (near-)zero ...
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49 views

How to deal with x/|x| in an equation?

How do I solve the following for x? $$ 0 = x-b+\lambda\frac{x}{|x|} $$ I'm trying to minimize $$f(x) = \frac{1}{2}(x-b)^2 + \lambda|x|$$ I took the derivative and now I'm trying to set it to $0$ and ...
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31 views

Minimum value of $F(a,b)$.

Let $$F(a,b) = \sum_{i=1}^n \left[ y_i - (ax_i+b) \right]^2$$ Find the minimum of $F$. Evaluating the dirctional derivatives: $$\frac{dF}{da} = \sum_{n=1}^n 2\cdot (y_i - (ax_i+b))(-x_i) \\ ...
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21 views

What is the solution to this maximization/minimization problem.

How would i go about solving the problem where i want to find $$max/min \quad \sum X_i$$ Where i have constraints which are $$M = P_1 X_1 + P_2 X_2+ P_3 X_3$$ and $$C_1 \geq X_1$$ $$C_2 \geq X_2$$ ...
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strong convexity of loss function in multi-dimensional (high-dimensional) space

My question is based on this paper (see the last 10 rows in page 7). It seems this is a general claim: In machine learning or statistic, the loss function $l(W^TX, y)$ (a linear predictor) can never ...
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Assigning people to events & events to time and place

I'm helping organizing a one-day conference which will host 6 events and a few hundred people, and wouldn't mind some help myself. There are four locations we've staked out with varying capacities, ...
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28 views

Maximization/Minimization under Kronecker product vectors with +semi definite constraint

I need to solve following optimization problems for $\mathbf{x}$ and $\mathbf{y}$ \begin{equation} \begin{array}{c} \text{max} \hspace{4mm} (\mathbf{x}\otimes \mathbf{y})^TA(\mathbf{x}\otimes ...
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1answer
16 views

How to derive this solution to this minimization problem in vector form?

We want to minimize the mean squared error $$ \sum_{t=1}^n (y_t - \theta^T x_t - \theta_0)^2. $$ Letting $X = [x_t, 1]$, we can rewrite the above problem in vector form as $$ \sum_{t=1}^n (y_t - ...
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How to find the extrema of a function?

I'm having problems finding the extrema of the function $h(x, y) = 2x sin(y) + y^2−x^2$. There is supposed to be one saddle point but I can't seem to get that. I tried taking $f_{xx}f_{yy} - ...
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Fish Eggs, Calculus, and Optimization

The survival of a fish egg through its critical period is a function of its mass, $x$. The larger the egg, the more nutrients are present and the more likely it is to hatch successfully. This ...
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37 views

Understanding a quadratic optimization problem

I'm relatively new to this kind of problem since I never faced a quadratic optimization problem before. The following question builds on a previously question of mine which has been kindly answered ...
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incremental approach to solve positive least square problem

Is there any incremental (approximate) solution for the following positive least squares problem: $$\min_x \|Ax-b\|^2\qquad \textrm{s.t.}\qquad x_i> 0,~b_1=1,~b_{i>1}=0$$
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Two quadratic programming problems always same answer?

Was exploring quadratic programming optimization and for two types of problems the answers seemed to always equal. Is there an intuitive proof? Problem 1: Minimize $\tfrac{1}{2} \mathbf{x}^T ...
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24 views

Search Direction in Conjugate Gradient

Could you help me with a Conjugate Gradient question? In using CG to solve $Ax = b$, why is the search direction $p_{k+1}$ in CG chosen as a linear combination of the residual $r_k$ and previous ...
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19 views

Could someone point me in the right direction for this complex analysis equation?

I'm supposed to show that the maximum value of $|z^2+1|$ on the unit disk $|z|\leq1$ is 2. My teacher's hint was "triangle inequality". I've been racking my brain how to tie the triangle inequality ...
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Metric Entropy Upper Bounds

In the paper Information-Theoretic Determination of Minimax Rates of Convergence the authors present Theorem 3 as follows: If $M_2(\epsilon)$ is the $\ell_2$ packing entropy of a density class ...
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How do you calculate Up and Down Penalties on a Branch and Bound algorithm of a MILP?

My notes really don't explain this clearly at all, so I have no idea what to do. If I have the following MILP: In which I've been told to solve it using: (a) Rule 1 (choose the variable with the ...
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What decides the structure of the dual variables taken in designing min-max type combinatorial optimization algorithms?

There are a bunch of combinatorial optimization problems like min cost flows and min weight perfect matchings that invoke duality and complimentary slackness to improve the primal feasible solution. ...
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58 views

No free lunch theorem

How does the No free lunch theorem apply in linear programming? Given a linear Programm. Calculate the optimal solution. Then you can calculate with the simplex method the solution in finite steps. ...
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Duality of Linear Program

Formulate the dual of the following linear problem: Min $\sum$ $\sum$ $c_{ij}$ $x_{ij}$ s.to $\sum$ $x_{ij}$ - $\sum$ $x_{ji}$ = 0, $\hspace{5mm}$ $x_{ij}$ >= $l_{ij}$ and $x_{ij}$ $\leq$ $u_{ij}$ ...
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Finding the Optimum Point on a Curve

I am trying to find the optimum point on a curve. More specifically the function of the curve I am looking at is: $f(x)=e^{0.3*ln(x+1)}$ and the curve looks like this: As I read in an old ...
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23 views

under what conditions the following matrix optimization has a unique solution?

So the problem is simple: Consider the following matrix optimization problem on matrix D. What conditions on the matrix dimensions should apply so that the solution to the minimum is unique. please ...
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Optimization - economics application, density functions and optimal labour supply

Assume and individuals preferences over consumption (c) and leisure ($l$) are described by the function: $u_i=c_i-a(b-l_i)^2$ The government provides lump-sum transfers to its citizens ($f$) ...
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Optimization (economics application) - Optimal tax rate and levels of education and housing

Every family has a preference over education (E) and housing (H) defined by the function: $U(E,H) = E^\alpha H^{1-\alpha}$ Households differ only with respect to income where household i's income is ...
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Optimal design for constrained Bayesian slope intercept model

Here is a problem I've been stuck on for quite a while. Consider the model \begin{equation} \mathbf{y}=\mathbf{H}\pmb{ \theta }+\pmb{\epsilon }. \end{equation} The design matrix is given by: ...
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Labeling constraints in a MILP

A manufacturing company consisting of two plants intends to introduce up to three new products. The production quantity of each product can be any number, integer or non-integer, but there is an upper ...
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Represent if-else or OR condition in a linear equation (optimisation with simplex algorithm)

I would like to write some linear equations and inequations to state that the sum of all possive x - C is smaller than L. As my ...
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Difference between Newton's method and Gauss-Newton method

I know that the Gauss-Newton method is essentially Newton's method with the modification that the Gauss-Newton method it uses the approximation $2J^TJ$ (where $J$ is the Jacobian matrix) for the ...
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2answers
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Implicit function theorem in comparative static problem

The individual lives for two periods. He has a utility function $u(c_{1} )+ bu(c_2)$. His budget constraint requires that his period I consumption be his period I endowment minus any savings, $c_1 = ...
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Find the minimum of $f(x) = x^2+\sin(x)$

I need to find the minimum of $x^2+\sin(x)$ but I can't get an answer. So far I've done this: The first derivative is $f'(x)=\cos(x) + 2x=0$ and the second derivative $f''(x)=-\sin(x) +2$ From the ...
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Lagrange Multipliers for linear functionals

Say I have a Banach-space $X$ and linear (!) functionals $f,g$, and I'm trying to solve the constrained optimization problem $$max~f(x)\quad s.t.~g(x)= 0,~\Vert x\Vert\le 1.$$ Suppose I can show ...
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1answer
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How to set up Lagrangian optimization with matrix constrains

Suppose we have a function $f: \mathbb{R} \to \mathbb{R} $ which we want to optimize subject to some constraint $g(x) \le c$ where $g:\mathbb{R} \to \mathbb{R} $ What we do is that we can set up a ...
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Modified bin packing problem

What is known: There are $b_i$ boxes of type $i$, where $i=1, \ldots, n$. There are $3$ types of truck, each can carry at most $K_1$, $K_2$, and $K_3$ boxes respectively. Each type of truck ...
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Reverse spline interpolation

Say I have a number of sets $(x, y)$ for $x \in \{0, 1, \dots, 255\}$. I want to find the least number of points to reproduce the set with a certain accuracy using linear interpolation. What is the ...
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1answer
25 views

Solving linear functions with constraints.

I have a table of data, for maximum available items in a given time period. With a constraint on how many of each items I can take in total. When I have taken the allowed amount of an item I can ...
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Why this two SVM problems are equivalent?

Consider the two classification problem, our classifier is a hyperplane $w^T x+b=0$, where $w$ is the weight vector, $x$ is the input vector and $b$ is the bias. we have input $x_i\in ...
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2answers
39 views

Given change in proportions and assuming minimum movement and a direction, calculate minimum proportion moving in that direction

Let $x_1, \dots, x_n$ with $\sum x_i = 1$ be proportions of a discrete distribution. Suppose the distribution changes and let $y_1, \dots, y_n$ be the subsequent proportions (and so $\sum y_i = 1$ ...
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How to find the pdf and likelihood function for bernoulli gaussian model

[Paper - A computationally fast approach to maximum-likelihood deconvolution by Chong-Yung Chi, Jerry M. Mendel and Dan Hampson link presents an ...
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solver for non-convex matrix optimization with convex constraints

So here is the problem: $\max_{D} ~~ \|A+BD\|$ subject to $\|D\|<1$ (any norm you like) where matrices A and B are given. The cost function is evidently convex as well as the constraint, but ...
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Solution of a linearly constrained quadratic programming problem

What is the solution of the following optimization problem: \begin{align} &\min{\mathbf{p}^\mathrm{T} \mathbf{B} \mathbf{p}}\\ &\text{subject to}: \mathbf{0}\leq{\mathbf{p}}\leq \mathbf{1}. ...
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51 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
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1answer
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Does optimal solution always occur at a vertex?

Is it true that if LP $ \text{max} \{c^Tx \ | \ Ax \leq b \}$ has an optimal solution, then $\exists$ a vertex which is simultaneously an optimal solution for LP? I know this works for LP of a ...
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Maximizing a convex function outside a convex set?

I want to prove the following equality: \begin{equation*} \min_{x: x^2 \ge t - x} x^2 = \max_{0 \le \mu \le 1} \left( \mu t + \frac{\mu^2}{4 (\mu -1)} \right). \end{equation*} The objective function ...
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Optimization problem of two variable

Find two numbers $a$ and $b$ with $a \leq b$ such that $\int_a^b (6-x-x^2)dx$ has the largest value.
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Dual Decomposition with multiple coupling constraints

This is probably a a simple question, but have been stuck on this for a while and unable to figure out my issue from the standard Boyd/Vandhenbergen decomposition references. I am interested in dual ...
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55 views

Maximization under Kronecker product vectors

I need some hints to solve the optimization problem on $\mathbf{x}$ and $\mathbf{y}$ \begin{equation} \begin{array}{c} \text{max} \hspace{4mm} (\mathbf{x}\otimes \mathbf{y})^TA(\mathbf{x}\otimes ...