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.

learn more… | top users | synonyms (5)

1
vote
0answers
37 views

What is the Euler Lagrange condition for SDEs?

Does the Euler Lagrange condition... $$\frac{d}{dt}\left( \frac{\partial L}{\partial \dot{x}}\right)-\frac{\partial L}{\partial x}=0$$ ...have a meaningful extension to Stochastic Differential ...
1
vote
0answers
146 views

Derivation of Von Karman Equations

I'm reading Howell's Applied Solid Mechanics to gain background for a research project. I'm struggling with the following derivation in the text that the authors refer to as a "lengthy exercise." The ...
3
votes
0answers
47 views

What does it mean for a problem to be time-homogenous?

(This is an associated question to Scaling in utility maximisation. $c_t$, $w_t$, $n_t$, $A$ are defined there.) I am reading that because of time homogeneity $$\sup_{(n,c)\in A(w)}E\left(\int_t^\...
0
votes
0answers
282 views

How to find minimum distance path between 2 points on a surface

Given a surface equation is $z=f(x,y)$ and also given two points on surface are $A(x_1,y_1,z_1)$ and $B(x_2,y_2,z_2)$ How can be found the path equation $(x=p(t),y=s(t),z=u(t))$ that it creates ...
1
vote
1answer
82 views

Minimize $w=9y_1+4y_2$ subject to linear inequalities

Minimize $w=9y_1+4y_2$ subject to : $4y_1+9y_2\geq 360$ $y_1+4y_2\geq 40$ $y_1\geq 0,~y_2\geq 0$
1
vote
1answer
16 views

Dual variable calculus

If $z=v'(w)$ and we introduce new variable $J(z)=v(w)-wz$. Then it is clear that $J'(z)=-w$ but why is $J''(z)=-1/v''(w)$?
1
vote
1answer
44 views

Scaling in utility maximisation

If I have the wealth process $$dw_t=rw_tdt+n_tS_t(\sigma dB_t+(\mu-r)dt)-c_tdt,$$ where $n$ is number of $S_t$ and $B_t$ is Brownian motion. If we define the admissible set $A$ as follows: $(n_t,c_t)\...
3
votes
2answers
90 views

Strategy for selecting the optimal time to check a cooldown timer

This is a hard problem for me to word in the title, so I'll try to do better now. Consider the following "game": You are sitting in a room beside a table. In the middle of the table there exists a ...
3
votes
0answers
230 views

Shortest system of roads between 4 cities

You have $4$ cities placed on the vertices of a square of side length $1$ km. You have to come up with a system of roads such that you can reach any city from another (directly or through another city,...
0
votes
0answers
34 views

Optimum point of $f(s) = \int_0^{\pi} \frac{ \exp(-s) y \cos(ky)}{s^2+y^2} \,dy $

Is it possible to find optimum point for the following function f(s) (i.e. $df/ds=0$): $$ f(s) = s e^{-s} \int_0^{a} \frac{ y \cos(\frac{\pi}{a} y)}{s^2+y^2} \,dy $$ or $$ f(s) = s e^{-s} \int_0^{a/...
3
votes
2answers
217 views

Finding an optimal sequence

It's my first time on this site:) I have to find a strictly increasing finite sequence $\{x_k\} _{k=1, \dots, n}$ with $x_1=c^2$ that will minimize the following expression $$\sum_{i=1}^n\sqrt{x_{i+1}-...
1
vote
1answer
64 views

Finding the smallest square inside a parabola. [duplicate]

I just thought of a problem earlier today, but wanted to know if there was an easier way of acquiring the answer. Say I have a standard parabola $y=x^2$ with 3 points on it $P,Q,R$ and another point $...
1
vote
0answers
20 views

Linear diophantine inequality maximum

For an irrational $\xi$ and given bound $x$, find integer $a, b \ge 0$ maximizing $y = a + b\xi$ subject to $y \le x$. $\xi$ is a square root of an integer, but I guess it doesn't matter. It's ...
3
votes
0answers
162 views

Jackson's theorem to optimize mean queue length of a traffic model

I am working on traffic signals for a city transport system. I modeled the city transport using a queuing network as shown in the following image Arrival rate of "A" cars from outside is S1 and ...
5
votes
1answer
78 views

Polynomial between $0$ and $1$ that produces largest integral

Question: Let $n\in \mathbb{N}$. Find the polynomial $p(x) = \sum_{i=1}^n a_ix^i$ that satisfies $p(1) = 1$ (and $p(0)=0$ since we already have $a_0=0$) $p(x) \in [0,1]$ for all $x\in [...
1
vote
0answers
48 views

Minimizing a Concave Function over a convex set

Here is the optimization problem that I am trying to solve. Thanks in advance for all help/insight provided. Let $T:[-2,2]^N\to\mathbb{R}_{-}$ be a concave function of its arguments. Given $\mathrm{y}...
3
votes
3answers
400 views

Determining the max area of a trapezoid with no known sides

A trapezoid is drawn inside a semi circle cross section with the upper base length, being the length of the circle diameter, $d$, and a lower base, $x$, touching the lower sides of the semi circle. ...
2
votes
0answers
51 views

Solving a nonlinear constrained optimization involving CDF and expectation of normal distribution

I would like to know if it is possible to solve the following nonlinear constrained optimization problem and find how the optimal solution varies with $C$ and $\beta$: $\max_{x,y}\beta F(x)-d(2-F(x)-...
0
votes
2answers
44 views

Optimization problem: Calculus 1

A company manufactures and sells $x$ units of a product per week. The weekly average cost in dollars per unit is $C =\frac13 x^2 + 9x + 17 + \frac{1552}{x}$ and the selling price in dollars per unit ...
0
votes
1answer
45 views

minimising quadratic function subject to integer solutions

I would appreciate if one could help me to solve this problem. I have a bivariate quadratic function: $$ f(a_1,a_2)=(1-u_1^2)a_1^2 +(1-u_2^2)a_2^2 -2u_1u_2a_1a_2 $$ where $u_1^2+u_2^2=1$ and $a_1$ ...
1
vote
0answers
101 views

Exercise about max and min of a 2D function with absolute value

I haven't done an exercise like this so, please, tell me if the proceeding is wrong and any kind of observations that you think can help me. Find global max and min of $$f(x,y)=|x^2-y|$$ in ...
1
vote
0answers
70 views

Why covariance constraint subsumes the average power constraint?

I am studying an optimization problem in the form of \begin{equation} \begin{aligned} &\underset{p(x)}{\text{maximize}} & & W\\ & \text{subject to} & & 0 \preceq K_{X} \...
2
votes
2answers
120 views

Finding extreme values of a variable on an intersection of a sphere and a plane

Determine the minimum and maximum value of the variable $z$ defined by the curve given by: \begin{cases} x^2+y^2+z^2=1 \\ x+2y+2z=0 \end{cases} So do I need to find a function $z=f(x,y)$ or just find,...
1
vote
0answers
34 views

Gradient w.r.t. boundary conditions in PDE

I am trying to solve the following problem. Suppose I have a field $\Phi(r)$, which is the solution to a partial differential equation: $\mathcal{L}\Phi(r) = s(r)$, as long as $r \neq r_0$ Here $\...
7
votes
1answer
175 views

What function maximizes area for a constant arc length?

Suppose I have a continuous function $f$, such that $f(0) = f(1) = 0$. Given the length $l$ of the curve between $0$ and $1$, which function maximizes the area under the curve? I know that if $l \leq \...
0
votes
1answer
24 views

maximum of function in bounded area

How can i calculate maximum of $ \frac{-1}{(x+y+3)^{2}} $ in [-1 1]x[-1 1] with non numeric method. I know that -0.2 is maximum of this function with numeric method and The Hesian matrix is zero . ...
2
votes
0answers
63 views

Hölder's inequality/Cauchy-Schwarz for Bregman Divergence?

Consider the Bregman divergence. $$ D_F(p, q) = F(p)-F(q)-\langle \nabla F(q), p-q\rangle. $$ And its dual norm: $D_{F*}(p, q) $ where $ F^*(y) = \arg\min_x \left\{ \langle x, y\rangle - F(x) \...
0
votes
0answers
49 views

Prove that $\int_{0 \le u \le 1,\Omega}g^2(x)udx$ in term of $u$ is convex

I am having a cost function and I want to know whether convex or not. Could you explain help me my problem? My problem is that given a cost function such as $$F(u)=\int_{0 \le u(x) \le 1,\Omega}g^2(x)...
2
votes
2answers
55 views

Find the maximum and minimum of the function $f$

Find the maximum and minimum of $f(x, y)=xy-y+x-1$ at the set $x^2+y^2\leq 2$. I have done the following: Since the region $x^2+y^2\leq 2$ is closed, $f$ has a maximum and a minimum, which is ...
3
votes
1answer
273 views

Proof of convergence for the proximal point algorithm

I'm trying to come up with a super simple proof of convergence on the proximal point algorithm, which uses the iteration scheme $x^{i+1} = \mathbf{prox}_{tf}(x^i)$ where $f$ is a closed, convex (...
-1
votes
1answer
147 views

How to hedge a sports bet

Suppose I've got a $200 ticket on the Golden State Warriors to win the NBA Finals at 5 : 1. The finals start next week, with the Cavs listed at 2 : 1 to beat the Warriors and the Warriors 4 : 9 to ...
2
votes
1answer
96 views

Algorithm - Maximum subarrays with sum and OR

I was thinking on the following problem: Given an array A. The value of an interval from i to the index j is defined as follows: Take the maximum value from that interval, and add it to the OR ...
2
votes
1answer
139 views

What would be the objective functions for this problem?

I have the following data (this is just a sample of my entire dataset): # Distance PriceIndex Rating 1 400 3 5 2 420 2 4 3 500 1 2 Considering the ...
0
votes
0answers
34 views

Find the extremas of the fuction $f$

I have to find the extremas of $f(x, y)=3x+2y$ subject to $2x^2+3y^2 \leq 3$. Since the region $2x^2+3y^2 \leq 3$ is closed, $f$ has a maximum and a minimum, which is either at the boundary or at ...
-1
votes
1answer
64 views

Theorem of Lagrange multipliers - Extremas of $f$ [duplicate]

I have to find the extremas of $f(x, y, z)=x+y+z$ subject to $x^2-y^2=1$, $2x+z=1$. I have done the following: We will use the theorem of Lagrange multipliers. The constraints are $$g_1(x,y,z)=x^...
2
votes
0answers
102 views

What subjects properly belong in operations research as their “owning” discipline?

Warning: This is a soft question, hence I would make it a wiki-community post if I could. Operations Research involves a broad swath of disciplines, ranging from probability and statistics/stochastic ...
2
votes
1answer
31 views

Shortest Path with Constraint

What is the length of the shortest path that goes from $(0,2)$ to $(12,1)$ that touches the $x$-axis? I tried using calculus to solve this problem (i.e.: distance is: $$ \sqrt{(x-0)^2 + (0-2)^2} + \...
3
votes
1answer
190 views

An Extension to the Generalized Eigenvalue Problem

Given two square matrices $A,B \in \mathbb{R}^{n\times n}$, the generalized eigenvalue problem is finding the scalar $\lambda \in \mathbb{C}$ and vector $x \in \mathbb{C}^{n}$ such that $$ A x=\...
0
votes
0answers
33 views

Maximising an area

I was wondering if someone could possibly explain this question: "A stadium should be oblong on plan with straight sides of length h and semi-circular arcs of radius r at either end. The facade must ...
1
vote
0answers
19 views

An equality between maximums of two logdet expressions

I have the following question. Let $K$ be a positive-definite $N\times N$ real-valued matrix (I'll denote this by $0\prec K$ and will subsequently assume all matrices are $N\times N$ and real-valued) ...
1
vote
1answer
48 views

Application Farkas Lemma

Let $A$ be a $m \times n$ matrix and $C$ a $k \times n$ matrix. Let $b \in \mathbb{R}^m$ and $d \in \mathbb{R}^k$. Show that exactly one of the following holds: a) There exists an $x \in \mathbb{R}^n$...
2
votes
2answers
67 views

What is the interpretation of the following optimization problem?

Suppose we have $N$ variables $x_1,\ldots,x_N$. Let $\mathbf{A}$ a $M \times N$ matrix, and $\mathbf{b}$ a $M \times 1$ vector. I have the following minimization problem: \begin{array}{rl} \min \...
5
votes
1answer
71 views

Is $\det(U_1\Lambda_1 U_1^t +U_2\Lambda_2 U_2^t +I)\le \det(\Lambda_1 +\Lambda_2 +I)$ correct?

I want to simplify or find an upper bound for the determinant $|K_1+K_2+I|$ where $I$ is identity matrix, $K_1$ and $K_2$ are positive semi-definite matrices of size $n$ and thus can be written as $...
2
votes
0answers
165 views

Lagrange multiplier expression

I would like to solve the following optimization problem using the gradient ascend method: \begin{array}{ll} \text{maximize}_{\theta} & \theta^TQ_1\theta + b_1\\ \text{subject to} & \theta^...
1
vote
0answers
20 views

How do we derive efficiency from robustness in the virtual ant solution to the traveling salesman problem?

Using virtual ants/swarm intelligence to solve the Traveling Salesman Problem is an example of using a robust system to solve an efficiency problem. We normally think of robustness and efficiency as ...
2
votes
2answers
154 views

Maximum perimeter for triangle inscribed in circle

How to prove that isosceles triangle has maximum perimeter from all trangles inscribed in circle? I found that from all isosceles trinagles - equilateral has maximum perimeter: Maximum perimeter of ...
2
votes
0answers
27 views

Optimization on Stiefel Manifold

$$\text{Find}~~U, V$$ $$\text{to maximize}~~f(U,V)=\text{tr}(U^TAVN)$$ $$\text{subject}~~U^TU=I_p,V^TV=I_p$$ where $N=\text{diag}(\mu_1,\cdots,\mu_p)$ with $\mu_1>\mu_2>\cdots>\mu_p>0$. I ...
1
vote
3answers
49 views

Constrainted optimization involving logarithms

The problem is to minimize $ f(x_1, x_2 ,x_3, x_4):= - \Big[ \log ({\frac{1}{4} + x_1}) + \log ({\frac{1}{2} + x_2})+ \log ({\frac{1}{5} + x_3})+ \log ({\frac{3}{4} + x_4}) \big]$ such that $...
9
votes
6answers
3k views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
3
votes
1answer
95 views

Hessian-Matrix positive definite $\iff$ $a$ local minimum?

It is commonly known that if $f$ is twice differentiable, $\nabla f(a) = 0$ and $H_f(a)$ positive definite, $a$ is a local minimum. So, in short: $H_f(a)$ positive definite $ \implies $ $a$ local ...