# optimization's questions - English 1answer

11.787 optimization questions.

### Binomial inequality

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 ...

### Prove that optimal solution is an extreme point in LPP.

1 answers, 1.136 views optimization linear-programming
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 ...

### Finding the gradient of a variant of a total variation regularized least squares cost function

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 ...

### 1 Sacred Geometry of Chance

1 answers, 105 views probability geometry optimization
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 ...

### What is the demand function p(x)?

1 answers, 1.007 views calculus optimization
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 ...

### 55 The most effective windshield-wiper setup. (Packing a square with sectors)

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 ...

### What is the computation time of LU-, Cholesky and QR-decomposition?

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 ...

### 3 does it make sense to talk about third-order (or higher order) optimization methods?

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. ...

### Properties of norms of inverse Hessians near a solution

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 ...

### 1 Optimization problem where the support of a random variable depends on the value of a decision variable

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

### 7 Prove $\min \left(a+b+\frac1a+\frac1b \right) = 3\sqrt{2}\:$ given $a^2+b^2=1$

8 answers, 200 views inequality optimization

### 2 Need to create $3-4$ different box sizes and to minimize material waste for a set of $n$ objects that need to fit into these boxes

1 answers, 55 views optimization operations-research
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 ...

### 2 Regularized Least Squares Using the Nuclear Norm

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

### 3 Regularized Least Squares - Generalized Tikhonov Regularization

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 ...

### 5 Shaking a box of rocks (Optimal Packing)

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 ...

### Constructing Any Arbitrary De Bruijn Sequence for Any Base Without Graphs

0 answers, 10 views combinatorics optimization
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 ...

### 2 How to search the feasible region in a smart way for the non closed-form optimization problem?

0 answers, 143 views optimization closed-form

### Does the multidimensional knapsack problem have the property of subadditivity?

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: $z_3 \leq z_1+z_2$ ?

### Can the solution of a linear program be irrational?

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 ...

### 1 Norm minimization for mechanical blind force identification?

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, ...