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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Find the maximum points of $f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$

Find the maximum points of $$f(x)=e^{-x}\sin^2(\pi x) \hspace{0.4cm},0<x<10$$ My calculations:I have calculated $f'(x)=\pi e^{-x}\sin(2\pi x)-e^{-x}\sin^2(\pi x)$ $f''(x)=e^{-x}\sin^2(\pi ...
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Monotonically decreasing function for multiplication product?

I have a set of numbers $S = [100,999]$ for which I want the maximum product $p$ such that $p = a \times b$ for all $a,b \in S$ also fulfilling some condition $C$. I would like $p$ to be monotonically ...
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Is there a two-dimensional method to optimally allocate N sampling points on a continuous function with derivatives?

I am looking for a method to optimally allocate sampling points. I have read some papers on this topic that discuss one-dimensional allocation using chebyshev points, but I haven't found a good ...
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4answers
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Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$

Is $\max\left(\frac{A}{B},\frac{C}{D}\right) \ge \frac{\max(A,C)}{\max(B,D)}$? Given that $A,B,C,D>0$. What about $\frac{A}{B},\frac{C}{D}>1$. Is there a better bound for the left ...
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1answer
22 views

Logistic regression maximum likelihood derivation

the following equations are given: $\sum_{j=1}^c\hat{P}_j = 1$ $\sigma_i(\mathbf{z}; \theta) = \frac{exp(\mathbf{\theta}_i^T\mathbf{z})}{\sum_{j=1}^cexp(\mathbf{\theta}_j^T\mathbf{z})}$ $L = ...
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24 views

Finding an optimal path for minimizing an integral.

Let $x,y$ be real numbers. Let the function $f(x,y)$ be real-entire in both $x$ and $y$. Thus $f(x,y)$ is a real-entire Taylor series in the variables $x,y$. How the find a non-intersecting path ...
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16 views

Show running time of algorithm on input of size n is $\Omega$ (f(n))

Basically I'm given this algorithm where I have an array A of integers which outputs an n-by-n array B where B[i,j] contains the sum of the array entries A and asked to give a bound of the form ...
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1answer
17 views

Derivation of Efficient Frontier (portfolio optimization) question

In Robert Merton's derivation of the efficient frontier of a portfolio, he minimizes $\frac{1}{2}\sigma^2 $ over the investment weights in each asset, where $\sigma^2$ represents portfolio variance. ...
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31 views

Lagrange Multipliers for Implicit Functions

How can I find the minimum / maximum of a function for one variable defined implicitly (f(x, y, z) = c) with a constraint g(x, y) = c on the domain? For example, say you wanted to minimize for z: ...
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maximizing a function involving factorial.

Can someone suggest a way to calculate the maximum with respect to $x \ge 1$ of: $$f(x)=\frac{1}{x!} \frac{1}{1-c^{1/\binom{x+n-1}{n-1}}}.$$ The constants $c$ and $n$ are parameters such that $c \in ...
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1answer
32 views

Designing linear systems to respond to particular kinds of oscillations

Say that I have a linear system which is being perturbed by an oscillating signal of a single frequency, of the form $$ \dot{\vec{x}}(t) = A\vec{x} + B \sin(\alpha t), $$ where $B$ is a vector of 1s ...
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10 views

confused between convex and non-linear optimziation

I have an optimization function which contains an objective function which contains sum of decision variables, division of sum of decision variables and also product of sum of decision variables. The ...
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2answers
18 views

How fast can you split a set of numbers into 2 sets, where the difference of each sum is maximized

How fast can you perform this task? More specifically, if there is a set of 2n elements, how fast could you split those elements into two groups of n elements where the sum of each group is of ...
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1answer
17 views

For single variable, being $x^*$ a local minimzer, suppose $|x-x^*|=\epsilon$. Find bounds on $|f(x)-f(x^*)|$ and $|f'(x)-f'(x^*)|$.

Im studying for a test on unconstrained optimization and completing exercises from a book that doesn't give the solution to all of them. This is one of them, I aren't sure if I am going the right way: ...
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maximum likelihood of a dirichlet prior

Suppose $\theta \sim D(\alpha)$ where $D$ denotes the Dirichlet distribution and $\alpha = (\alpha_1,\ldots,\alpha_K)$ its hyperparameter, in which case: $$p(\theta) = \frac{\Gamma(\sum_k ...
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2answers
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Is $[u_1,u_2]$ an edge of the polytope $conv(F)$?

Here's the problem. I have a finite set of vectors $F \subset \mathbb{Z}_{\geq 0}^d$. I define $P$ to be the convex polytope $conv(F)$ i.e. the convex hull of $F$. Given $u_1, u_2 \in F$, is the ...
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44 views

Integer Optimization

I have an integer optimization problem that I've been pondering for the last several days. Here's an abbreviated version: I have several wav song files with variable sizes (601201 kilobytes for ...
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41 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
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13 views

Equivalence of Two Statements (Duality Theory, Optimization)

Let $a$ and $a_{1}, ... , a_{m}$ be given vectors in $\mathbb{R^{n}}$. Prove that the following two statements are equivalent. $a)$ For all $x \geq 0$ we have $a'x \leq max_{i} a_{i}'x$. $b)$ There ...
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25 views

optimisation problem with linear constraint

optimisation problem with linear constraint I have an optimisation problem. I wish to maximise a function subject to a constraint. It is the constraint that is causing me problems. I am using an ...
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23 views

Quick help on showing a set is bounded

I am working on a constraint optimization problem. I have found the extrema and all I need to do now is to show that the set S that the critical points are defined in is bounded and closed (therefore ...
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24 views

Properties determining boundedness of function

The function I am looking at is $$f(x) = \frac{1}{2}x^TAx + b^Tx + c$$ where $A$ is a symmetric matrix in $\mathbb{R}^{n\times n}$ and $b,c$ belong to $\mathbb{R}^n$ I want to determine what ...
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How to solve the following minimization problem? [duplicate]

For $n$ scalars $a_1,...,a_n$, find the solution of $$\min_{x \in \mathbb R}\sum^n_{i=1} |x - a_i|$$ I denoted $$\delta(x) = \sum^n_{i=1} |x - a_i| $$ and found $$ \delta'(x) = \sum^n_{i=1} ...
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1answer
16 views

partial derivative of a vector with respect to a variable

I have a vector in the following form $\mathbf{w}^T = [a_1*w_1, a_2*w_2, \dots, a_d*w_d]$ what is the partial derivative of $\mathbf{w}$ with respect to $w_j$ ? (1 or 2) $\frac{\partial ...
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Minimizing 1-dim problem containing quadratic and sum of absolute value functions

I stumbled on a problem which I am not sure how to most efficiently solve - I want a solver in code which I need to repeat several times with various constants. Basically I want to minimize a 1-dim ...
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25 views

How to prove that a function is compact (closed and bounded)?

The specific function I am looking at is $f(x_1,x_2) = x_1x_2 + \frac 1{x_1} + \frac 1{x_2}$, where for a fixed $a > 0, f(x) \le a$ and $(x1,x2) > 0 $ I'm really just looking for where to ...
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The relation between two different definitions of Affine sets

I am following a presentation, which says that for an affine set $L \subseteq \mathbb{R}^n$ it is: $$L=\left\{x|Ax=b \right\}$$ for some $A,b$. The first definition of $L$ as an affine set is given ...
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Separate a list of spheres into several lists, each contained in a sphere with a radius no larger than specified.

I have a list of arbitrary spheres, what I want to end up with is that list separated into a number of groups, where spheres in each group all fit into thier specific larger sphere. The limitation is, ...
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Calculus optimization word problem

"A printed page is being designed to contain up to 96 square inches of printed material. The margins are 1” on the left and right and 1.5” on the top and bottom. Find the outer dimensions of the page ...
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Fenchel dual vs Lagrange dual

Consider the Fenchel dual and the Lagrangian dual. Are these duals equivalent? In other words, is using one of the these duals (say for solving an optimization), would give the same answer as using ...
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How to reduce 3 dimensional optimization to 2 dimensions?

I am trying to minimize the surface area of a parallelepiped of unit volume. Using Volume = xyz(1 + 2cos(a)cos(b)cos(c) - cos^2(a) - cos^2(b) - cos^2(c))^1/2 = 1 where x,y,z are edge lengths and ...
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How to find the parallelepiped of unit volume with minimal surface area?

Is it best to approach this problem using edge lengths and the angles between them? I am trying to reduce the problem to two dimensions, although I haven't successfully done so yet So I have Volume ...
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Hyperplane optimization for Support Vector Machines

I am trying to learn about the theory behind the Support Vector Machines, by reading the tutorial at: http://research.microsoft.com/pubs/67119/svmtutorial.pdf In its most basic form, SVMs is a binary ...
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Approximating the max value of a function containing the half-sum of binomial series

I recently met a problem and I have been finding related materials about it. However it is still hard to handle. Precisely, I want to maximize a function related to $$S_n = \sum_{i=0}^{\lfloor ...
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1answer
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Coercive or not?

I had this problem in the exam. Let $X = [x_1,...,x_d]^T$, $a\space \epsilon$ $\mathbb{R}^d$ and $C$$\epsilon$$\mathbb{R}$. Argue for or against. $f(X) = a^TX + C||X||^2$ is coercive only for $C ...
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Extreme of a function

Let us have the following function on $x\in[0,1]$ $$ y =f(x)= x + a\left(\max(0,b-x)\right) $$ where $a>0$ and $b\in[0,1]$ are known parameters. Could you please find the solution of this $$ ...
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Least sum of distances

Problem: Let $A, B, C, D$ be points in a $3$-dimensional space. Find the point $X$ that minimizes the sum of the distances $AX+ BX + CX + DX$. Context: During a course, I was assigned a ...
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How to Extract the dual feasible search directions for the primal-dual potential reduction algorithm?

I am trying to implement the 4.4 Primal-dual potential reduction algorithm introduced in M.S Lobo et al.. Here is a screenshot depicts the algorithm flow: As ...
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Newton Raphson Method Overestimating Parameters

I have implemented an almost plain vanilla algorithm to find the MLE estimates of 3 parameters in a log-likelihood function (in R.) When I test my algorithm with some simulated data it does pretty ...
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Condition that multiplied hermitian matrix stays hermitian

Suppose we are given a hermitian matrix $E \in \mathbb{C}^{n\times n}$. I want to find sufficient conditions on the entries of a real symmetric matrix $M$ (depending on the entries of the given ...
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Solving Optimal Control with non linear cost function

I am trying to solve the Kermak Mc-Kendrick SIR model using a non linear cost function, but I am stuck on how to possibly solve it. I need to find an optimal control $u(t)$ in $[0,T]$ that minimize: ...
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Question regarding KKT conditions in optimization

Following is Proposition 3.3.7 in Bersekas' Nonlinear Programming. Let $x^*$ be the local minimum of the problem: $$\text{Minimize }\; f(x) $$ $$ \text{subject to: }\ h_j(x) = 0, ...
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Consider the problem minimize $f(x)= x^4 −1. $

I am studying for a test and I found this problem in the textbook that I'm using, there may be a conceptual problem that I'm coming across. My test is on unconstrained optimization (multi-variable) ...
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3answers
30 views

minimize expression

How can the following expression be minimized wrt w: $$ \frac{w^T D w}{w^T S w}, $$ where $w \in \mathbf{R}^n$, $D \in \mathbf{R}^{n \times n}$ symmetric, and $S \in \mathbf{R}^{n \times n}$ ...
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1answer
38 views

Minimize the Frobenius norm of the difference of two matrices with respect to matrix: $\underset{B} {\mathrm{argmin}} \left\| A- B \right\|_F$

The following question is similar to this one, but I think that it is not straightforward to move from one to the other, so please take a look. Otherwise, please let me know and I will delete it. ...
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How is called this an optimization problem of this kind, or which techniques could I use to solve it?

I have an optimization problem which is a multivariable problem(34 variables), I need to find the minimum cost but my solution must be only concerning to the value of 3 variables out of the 34; the ...
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1answer
28 views

Sufficient condition for global maximum of strictly quasi-concave functions (unconstrained)?

Suppose $f(x)$ is a function from $\mathbb{R}$ to $\mathbb{R}$ and $f$ is strictly quasi-concave. If $x^*$ is a point such that $f'(x^*)=0$, then can we say that $x^*$ is a global maximum of this ...
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Are there any standard methods to solve a linear objective with nonconvex constraints?

I see that nonlinear programming entails nonlinear objectives with convex or linear constraints. Is there any theory/method to solve linear objective with nonconvex constraints and some convex ...
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1answer
18 views

Multivariable optimization - how to parametrize a boundary?

A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by $T(x,y) = 2x^2 + y^2 - y + 3$. Find the ...
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Differentiability of Moreau-Yosida Regularization? [duplicate]

I'm looking for a proof of the differentiability of the Moreau-Yosida regularization of a proper closed convex function $f(y)$ defined on an n-dimensional Banach space $Y$. namely the function is ...