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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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 ...
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2answers
59 views

Image restoration in matlab via PDE toolbox

I want to remove a noise for an image using matlab, when the observed image is $$f=u+v$$ where $u$ is the restored image (is the image i want recovered) and $v$ is the gaussian noise. To restore $u$, ...
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59 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 ...
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Dropping Multiple Constraint in Active Set Optimization

Does anyone know an example for the Exercise 2.5.2 of Bertsekas nonlinear programming book?: Show by example that in the method of this section, if several constraints $a_j'X \le b_j$ with ...
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70 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 ...
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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 ...
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113 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. ...
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19 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 ...
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39 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 ...
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26 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 $a_1$ ...
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35 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 ...
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64 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} ...
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2answers
47 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 ...
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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 ...
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135 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 ...
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1answer
19 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 . ...
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23 views

Holder's inequality/Cauchy-Schwartz 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) ...
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44 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 ...
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64 views

Help with Linear Algebra Optimization Problem. 4 people crossing a bridge

"Four people, A, B C and D need to get cross a river. The only way to cross the river is by an old bridge, which holds at most 2 people at a time. Being in dark, they can not cross the bridge ...
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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 ...
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1answer
52 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 ...
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54 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 ...
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73 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 ...
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92 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 ...
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28 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 ...
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45 views

Theorem of Lagrange multipliers - Extremas of $f$

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 ...
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32 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 ...
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23 views

Linear Algebra L2 minimization

Im really confused about how to solve this question or even what its asking. Any help would be much appreciated! Let A be an m x n real matrix ($m \gt n$). Let x* be the minimizer of $||Ax - b||^2 + ...
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26 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} + ...
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102 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 ...
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27 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 ...
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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) ...
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30 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 ...
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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 ...
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How to distinguish different states?

The linear model for sparse coding is y = Ax, where x is a sparse vector (m*1), A is the dictionary matrix (n*m), and y is the observation vector (n*1). If ...
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61 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 ...
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How to solve this optimization problem (Interpolation on trigram, bigram, and unigram for language model)?

I am a newbie in optimization and learn about the language model in NLP. I am studying the basic interpolation method to estimate the probability of the current word given the last 2 words, $P(w_i | ...
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150 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} & ...
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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 ...
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151 views

Linear - Quadratic optimization for system of objectives

I have two distinct data sets, $\{x^{\mu},J^{\mu}\}$, $\mu=1,\ldots,n$ and $\{x^{\nu},V^{\nu}\}$, $\nu=1,\ldots,m$ that also include uncertainties $\delta J^{\mu}$ and $\delta V^{\nu}$. In these I fit ...
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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 ...
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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 ...
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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 ...
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Deriving Dual Averaging from (Sub)gradient Descent

Here the presenter tries to derive a simple Dual Averaging from (sub)gradient descent. I have a little problems understanding the steps. (Sub)gradient descent: Loop through: $$ x_{k+1} = x_k - t_k ...
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538 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?
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51 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 ...
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Dynamic programming approach for multidimensional problem

I use a dynamic programming approach to optimize the behaviour of individuals playing a game.I have one strategy matrix that describes the behaviour of individuals in situation 1, which depends on ...
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Functional Minimization of Exponential Decay

I would like to find a function $f$ that minimizes the functional: $$\ln(f(x))f(x)-\frac1x$$ over some range of $x > 0$. Is this a good application for functional calculus and the Euler-Lagrange ...
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Find extremum of functional

I want to find the extremum of $$J(y)= \int_1^2 \frac{\sqrt{1+y'^2}}{x}dx, \ y(1)=0, \ \ y(2)=1$$ I thought to use the following theorem: If $y$ is a local extremum for the functional $J(y)= ...
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49 views

Why does the functional have a local minimum at $0$?

Definition: Let $J: A \to \mathbb{R}$ be a functional , where $A \subset V$ and $(V, ||\cdot||)$ a linear space with norm. Let $y_0 \in A$ and $h \in V$ such that $y_0+ \epsilon h \in A $ for ...