**11.787 optimization questions.**

I have two binomial expressions.
\begin{align}
mm_{1}\binom{N(d+1)+n}{n}\tag{1}\\
\binom{N+n+1}{n+1}\tag{2}
\end{align}
wher $n,m,m_1,d$ are constants and only $N$ varies.
Now we have the ...

While proving this I have proved that Optimal solution cannot lie inside the feasible set and that each supporting hyperplane to a set bounded from below (which is the case as in LPP we can always set ...

As the question states, given a sum-function :
$$f(x) = \sum_{ij}\left({\sqrt{(x_{ij} - y_{ij})^2+1}}+\frac{1}{2}\sqrt{(x_{ij}-x_{i+1j})^2+(x_{ij}-x_{ij+1})^2 +1}\right)$$
where $x_{ij} $ describes ...

This problem is dedicated to Leon the professional.
first this question came to my mind when I was contemplating on the numbers 1-20 arranged interestingly around a regular dartboard, then I proposed ...

The marginal revenue of a certain commodity is $R^1(x)=-3x^2+4x+32$
where $x$ is the level of production in thousands. Assume $R(0)=0$ Find $R(x)$. What is the demand function of $p(x)$?
I took the ...

I was on the bus on the way to uni this morning and it was raining quite heavily. I was sitting right up near the front where I could see the window wipers doing their thing. It made me think "what is ...

I found these information about computation-time of following decompositions:
Cholesky: (1/3)*n^3 + O(n^2) --> So computation-time is O(n^3)
LU: 2*(n^3/3) --> So computation-time is O(n^3) also (not ...

In optimization, most methods I've seen are some variant of gradient descent (popular in machine learning) or methods that use information about second derivatives, ie, the Hessian matrix, with the ...

Let's say we have a functional $J[f] = \int_a^b L(x, f, f') dx$ that we are trying to minimize. I'm trying to think what is the best way to do this numerically, with something like gradient descent.
...

In theorem 3.5 of the book on Numerical Optimization by Jorge Nocedal and Stephen J Wright, Second edition, is given the proof for the quadratic convergence to the solution of a sequence of iterates ...

Suppose we have a dynamic program / Markov decision process where in the objective we have something like this: $v_t(s)=\max\limits_{u_1,u_2} \{r_t(s,u_1,u_2) + E[v_{t+1}(u_1+\min(u_2,\epsilon))]\}$, ...

Prove that
$$ \min\left(a+b+\frac1a+\frac1b\right) = 3\sqrt{2}$$
Given $$a^2+b^2=1 \quad(a,b \in \mathbb R^+)$$
Without using calculus.
$\mathbf {My Attempt}$
I tried the AM-GM, but this gives $\min ...

I have the following problem to solve
Let $\mu \geq 0$ and $\nu \geq 0$ be probability vectors (meaning $1^T\mu = 1^T\nu = 1)$. We have a set of matrices $\Pi(\mu, \nu)$ where for each $P$ we have $...

Excuse the non-mathematical way I've phrased the question.
I have the following problem:
I have $N$ square paper documents with side lengths between $150$mm and $860$mm. I know each document side's ...

I have the following least squares nuclear norm problem,
$$
\min_{\bf X} \frac{1}{2}{\left\lVert {\bf b} - {{\bf W}}vec({\bf X}) \right\rVert}^2_2 + {\lambda_*}\Arrowvert {\bf X} \Arrowvert_*
$$
...

In Image Restoration, a true image $f$ (in vector form) can be related to degraded data $y$ through a linear model of the form
$$y = Hf + n$$
where $H$ is a 2D blurring matrix and $n$ is a noise ...

My coworker was telling me that when he plants seeds on his farm, he puts them all in a large container on the tractor and after a period of just driving, the seeds are more densely packed than when ...

So I had been playing with De Bruijn Sequences/Cycles (a clycical sequence that lists every possible combination of digits of a certain length in a certain base with no repeats) before I even knew ...

A random variable X follows a discrete distribution D(P,I), where $P = \{p_1,p_2,...,p_n\}$ is the the list of the probability, and $I=\{i_1, i_2, \cdots, i_n\}$ is the list of values. $T=X_1+X_2+\...

Summary
This question is about covering a sphere with great circles so that the resulting density is uniform and that if we for each circle associate a normal vector, then the magnitude of the sum of ...

I am trying to solve the following minimization problem
\begin{equation*}
\begin{aligned}
& \underset{X,Y \in \mathbb{R}^{n\times k}}{\text{minimize}}
& & \| X Y^\top A - B \|_{\text F}^2 ...

Line search methods for convex optimization are of two main types
1) Exact line search - explicit minimization $\min_\eta f(x+\eta \,\Delta x) $
2) Inexact line search (Backtracking example) - Pick $...

If $z_1$ is the objective value over the input set $c_1$ and $z_2$ over $c_2$ and $z_3$ over the set union($c_1,c_2$) then: $z_3 \leq z_1+z_2$ ?

Let's consider a generic linear programming problem. Is it possible that the decision variables of the objective function assume (at the optimal solution) irrational values?
Also, is it possible that ...

The differential equations steering motion in classical mechanics are very famous:
$$\sum_i\frac{{\bf F}_i}{m}={\bf a} = \frac{d{\bf v}(t)}{dt} = \frac{d{\bf x}(t)^2}{dt^2}$$
Now to the question, ...

Let's simply say:
$f: \mathbb R^{2} \to \mathbb R, f(x,y):= xy$ and $h: \mathbb R^{2} \to \mathbb R, h(x,y):=x+y-1$
I am aware of the usual lagrange multiplier method, in other words:
Let $\lambda ...

I've learnt some ways of finding a point on a line which can minimize the sum of length to other points.
I want to generalize this to three points using geometric methods.
Question:
There're ...

Let $x$ be a real random variable with $\textbf{prob}(x=a_i) = p_i$, where $a_1 < a_2 < \cdots < a_n$. In this case, show whether $$\textbf{quart}(x)\leq \alpha$$ is convex in $p$ or not. (...

Please I have difficulty relating the minimax theorem to the Hahn Banach theorem in functional Analysis...minimax is a consequence of Hahn Banach but I just can't see it... Please I need someone who's ...

I am trying to solve the following convex minimization problem:
minimize: $f(x)$
subject to: $g(x)\leq 0.$
Where $f, g$ : $\mathbb{R}^{n}\rightarrow \mathbb{R}$ are convex functions (not ...

I have a maximization problem, with $n$ variables, $2$ equality constraints and $n$ inequalities. I tried to solve it using the simplex method, but that needs the same number of constraints as ...

Let $\alpha > 1$ and $s >0$
find all the local extrema of $f: \mathbb R_{\geq 0} \times \mathbb R_{\geq 0}\to\mathbb R, f(x,y):=x^{\alpha}+y^{\alpha}$
under the constraint that $h(x,y):=x+y-s=0$
...

I am studying the max-sum algorithm to solve Distributed Constraint Optimization Problem. I have a very basic doubt about the maximization of a function w.r.t. a single variable. Consider the ...

Let $X$ be a random variable with $X \in [a,b]$, known mean $\mathbb{E}[X] = \mu$ and known variance $\text{var}[X] = \sigma^2$. We are interested in an expression for the worst-case quantile. That is,...

I'm going to use scale-based preconditioning in a quadratic optimization problem: minimize $ x^T Q x + p^T x$ such that $ A x + b = 0$ and $D x + E \leq 0$, I want to speed up finding the optimal $x$ (...

In Boyd and Vandenberghe's textbook on Convex Optimization, is claimed that:
We always have $$ \underset{x,y}{\text{inf}\; f(x, y)} = \underset{x}{\text{inf}}\;\tilde{f}(x) $$
where $$ \tilde{f}(...

I think to let $b=[f(x_1) \dots f(x_m)]^T$ and $a=[\cos(x_1) \dots \cos(x_m)]^T$ s.t.
a
$$h(c) := \sum_{k=1}^{m} (f(x_k)-c \cos(x_k))^2 = (b-ca)^{T}(b-ca)$$
Then expanding and setting $h'(c)=0$, we ...

I have a collection of quadratic functions
\begin{align}
f_i(x) = \frac{1}{2}x^T Q_i x, \qquad i = 1,\dots,m,
\end{align}
where each $Q_i$ is an indefinite $n \times n$ matrix and $x \in \{-1,1\}^n$....

We are given the polyhedron $X=\{x:Ax\le b,x\ge 0\}$ and the point $y\in X$. We want to find a point $x \in X$ such that $d(x,y)$ is maximized. The function $d(x,y)$ represents the distance between ...

Given a discrete probability distribution with $n$ possible outcomes and the task of repeatedly sampling until getting each outcome at least once, does the uniform distribution give the smallest ...

Is there a theory of constrained optimization with complex variables, do you know any textbook on that topic? The typical textbooks on constrained optimization deal with real variables. I actually ...

I know that the Lagrange multiplier method helps us evaluate critical points of $f$ on the closed boundary of the restriction.
In other words we solve:$$\nabla f=\lambda \nabla g$$
But why does ...

Let's take a simple example $f: \mathbb R^{2} \to \mathbb R$, $f(x,y)=xy$ and then I want to treat $f$ for a constraint $M$ under all possible inequalities:
Case 1) $M:=\{(x,y)\in \mathbb R^{2}|x^2+y^...

I have a simple question. What's the difference in behaviour between saturation limit and constrained limit in control theory?
We say that we got this objective function:
$$J_{min} = \frac{1}{2}x^...

I have a $m\times n$ matrix $A$, such that $m>n$. Matrix $A$ is also comprised of only the values $-1,0,1$. We also have a vector $\vec b$ that is $m\times1$. $\vec b$ only has the restrictions ...

While minimizing a lipschitz continuous strongly convex functions, the rate of convergence of gradient descent method depends on the condition number of the hessian of the function where high ...

At first I thought the following problem looked simple, but I've had serious problems pinning it down:
Suppose that $\prod_{i=1}^k x_i$ is fixed. Then find the minimum value of
$$\sum_{i=1}^k (1 - ...

Let $M:=\{(x,y,z) \in \mathbb R^{3} : x^2+y^2+z^2\geq 36\}$ and $f: \mathbb R^{3} \to \mathbb R$, $f(x,y,z)=\frac{x^2}{2}+\frac{y^2}{4}+\frac{z^2}{6}$
Justify why $f|_{M}$ takes on a minimum
The ...

Consider a polynomial optimization problem of the following type
\begin{equation}
\begin{array}{cl}
\text{maximize} & p(\mathbf{x}) \\
\text{subject to} & \mathbf{x} \in K \\
\end{array}
\...

Dear Convex Analysis Experts,
Is $\left\{a \in \mathbb R^K \: | \: f(0)=1, \: \lvert f(x)\rvert \leq 1 \text{ for } \alpha \leq x \leq \beta\right\}$, where $f(x) = \sum_{i=0}^{K-1} a_i x^i$, ...

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- algorithms
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