Articles of optimization

Local optimality of a KKT point.

Consider the problem \begin{equation} \min_x f(x)~~~{\rm s.t.}~~~ g_i(x)\leq 0,~~i=1,\dots,I, \end{equation} where $x$ is the optimization parameter vector, $f(x)$ is the objective function and $g_i(x)$ is the $i$th constraint function. $I$ denotes the number of constraints. Consider now a point $x^\star$ that satisfies the Karush-Kuhn-Tucker conditions. In general, in non-convex optimization, a KKT point can be […]

The solution of $\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{\sin rx}\right|$

I am looking for the solution of $$\min_x \max_{1\leq r \leq N} \left|\frac{\sin rMx}{ M \sin rx}\right|$$ where $M < N$ are integers and $x \in \mathbb{R}^+$. For $M = 4, N = 6$, $f_{r,M}(x) =\left|\frac{\sin rMx}{ M \sin rx}\right|$ is plotted for different values of $r$ in the figure below. The maximum is plotted […]

Clarification on optimization problem

While reading a combinatorics paper about packing densities in compositions, I encountered the following optimization problem. Maximize $$f(\alpha_1, \dots, \alpha_5) = \sum_{1 \le i < j < k \le 5} \alpha_i \alpha_j \alpha_k$$ with the constraints $$0 \le \alpha_1, \dots, \alpha_5 \le 1, \quad\sum_{i=1}^5i\alpha_i=1$$ (See below for written out expressions.) The authors conclude that $\alpha_5 […]

How can we draw $14$ squares to obtain an $8 \times 8$ table divided into $64$ unit squares?

How can we draw $14$ squares to obtain an $8\times8$ table divided into $64$ unit squares? Notes: -The squares to be drawn can be of any size. -There will be no drawings outside the table.

Why these two problems lead to same answers?

Suppose these two problems: Problem 1: $$\min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {|\sum_{1\leq i\leq N_p}^{N_p}x_ie^{\frac{2\pi l}{N}p_i}| \over {\sum_{i=1}^{N_p} x_i^2}} \quad \equiv \quad \min_{X,P} \quad\max_{1\leq l\leq L-1} \quad {\sum_{1\leq i,j\leq N_p}^{N_p}x_ix_je^{\frac{2\pi l}{N}(p_i-p_j)} \over {(\sum_{i=1}^{N_p} x_i^2})^2} $$ Problem 2: $$\min_{P} \quad\max_{1\leq l\leq L-1} \quad |\sum_{1\leq i\leq N_p}^{N_p}e^{\frac{2\pi l}{N}p_i}| \quad \equiv \quad \min_{P} \quad\max_{1\leq l\leq L-1} \quad \sum_{1\leq i,j\leq […]

“Box With No Top” Optimization

I am having some trouble with this problem, A box with no top is to be constructed from a piece of cardboard of dimensions $A$ by $B$ by cutting out squares of length $h$ from the corners and folding up the sides as in the figure below. Suppose that the box height is $h = […]

Distribute small number of points on a disc

Firstly I strongly know how many similar questions there are here. It’s about sets of evenly distributed points inside a circle. If we need a big set of such points, good solutions are: Isocell method; Geogebra example Spirales method; Geogebra example But it’s interesting for me: if we have small number of points: 2,3,7,.. what […]

Maximising sum of sine/cosine functions

I have got a problem and I would appreciate if one could help. I have to maximise following function that is the sum of sine/cosine functions: $$ f(x,y)=a_1 \cos(x) +b_1 \sin(x)+ a_2 \cos(y) +b_2 \sin(y)+ a_3 \cos(y-x) +b_3 \sin(y-x) $$ and find the $x$ and $y$ values where the function $f(x,y)$ is maximised assuming that […]

Maximal area of a triangle

What would be the most elementary proof of the following: A triangle has been drawn inside the circle with radius $r$ and its area is as large as possible. Prove that the triangle is equilateral. I mean, can we avoid topology and compactness to prove that the maximal triangle can be found? And can we […]

Maximum area of a isosceles triangle in a circle with a radius r

As said in the title, I’m looking for the maximum area of a isosceles triangle in a circle with a radius $r$. I’ve split the isosceles triangle in two, and I solve for the area $A=\frac{bh}{2}$*. I have made my base $x$, and solve for the height by using the Pythagorean theorem of the smaller […]