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)

0
votes
1answer
20 views

How to solve a linear program with additional equality constraints?

The following optimization problem $$\max_{\substack{x \ge 0,\\Ax^T+b^T\ge 0}} c x^T$$ where $x \in \mathbb{R}^n$, $A \in \mathbb{R}^{m \times n}$, $b\in\mathbb{R}^m$, and $c \in \mathbb{R}^n$ is ...
0
votes
0answers
16 views

Distributed control problem which involves the p-Laplacian operator

Someone could help me to deduce the optimality system for the optimal control problem: \begin{align} &\min_{u\in L^{2}(\Omega)} ...
0
votes
0answers
15 views

Minimum number of $m \times m$ matrices needed to recover a single large matrix

This problem was motivated by the need to efficiently train a neural net on a dataset in which the labels represent dependencies between examples, but nothing about it is machine-learning specific so ...
0
votes
0answers
10 views

Different ways of solving $\underset{\mathbf{s}}{\text{min}}\;\|F\mathbf{s}-\mathbf{x}\|_{l_2}^2 + \|W\mathbf{s}\|_{l_2}^2$ least square problem?

The problem that I am going to describe arises from compressed sensing technique and after using weighted least squares it can be transformed into the following least squares problem: ...
0
votes
0answers
24 views

Drift management optimization

I have a problem in which I am having trouble formulating the optimization. A portfolio value is $10M I have a vector of current weights [.10,.15,.15,.10,.05,.10,.20,.15] and another vector of ...
0
votes
1answer
30 views

how I can minimize this equation using derivation

I'm a software engineer and have not much mathematical knowledge. Now, I'm facing with a problem in my research. I have a system of equations as below: $$P_1 = \alpha V_p + \beta I_c^2 $$ $$P_2 = ...
0
votes
2answers
1k views

How to calculate the hessian of a matrix function

I am estimating a model minimizing the following objective function, $ M(\theta) = (Z'G(\theta))'W(Z'G(\theta))$ $Z$ is an $N \times L$ matrix of data, and $W$ is an $L\times L$ weight matrix, ...
2
votes
1answer
66 views

Are there any global extrema in this Lagrange Multiplier problem?

I'm trying to find the max and mins of the equation $f(x,y,z) = xy + 3xz + 2yz$ on the constraint, $g(x,y,z)=5x+9y+z-10$. So according to the Lagrange Multiplier procedure, I take the partial ...
0
votes
2answers
758 views

Optimization, rectangle inscribed inside arch of the curve.

A rectangle is to be inscribed under the arch of the curve $y = 4\cos(0.5x)$ from $x = \pi$ to $x = -\pi$. What are the dimensions of the rectangle with largest area, and what is the largest area? ...
0
votes
0answers
10 views

Maximize function symbolically

I have the following expression: $$ \sum_{i,j=1}^n\rho_{ij}^2-\frac{2}{n}\sum_{i=1}^n\left(\sum_{j=1}^n\rho_{ij}\right)^2 +\frac{1}{n^2}\left(\sum_{i,j=1}^n\rho_{ij}\right)^2 $$ My goal is to ...
6
votes
1answer
67 views

Calculus optimization problem leads to a quartic polynomial - is there a better way?

I am tutoring a student in first-semester Calculus. He needs to minimize the function $$f(x)=\frac{\sqrt{4+x^2}}{2}+\frac{\sqrt{1+(3-x)^2}}{4}$$ Taking the derivative and setting it equal to zero, we ...
1
vote
2answers
31 views

Optimization Cost of candy

You have decided to buy candy for the trick-or-treaters and have estimated there will be 200 children coming to your door, and plan to give each children three pieces of candy. You have decided to ...
0
votes
0answers
11 views

generalised eigenvalue problem with absolute value

Problem: $\max_w |w^t A w|-|w^t B w|$ s.t $w^t C w=1$ If there was no absolute values, i.e. if the problem was $\max_w w^t A w-w^t B w$ s.t $w^t C w=1$ this would, by using the appropriate Lagrange ...
6
votes
2answers
294 views

The maximum value of $PA\cdot PB\cdot PC$

Let $A,B,C$ be the vertices of a triangle inscribed in a unit circle, and let $P$ be a point in the interior or on the sides of the $\triangle{ABC}$ .Then the maximum value of $PA\cdot PB\cdot PC$ ...
0
votes
0answers
7 views

Bipartite Matching with quadratic objective

I'm looking for the best way to formulate and solve the following bipartite matching problem: I have n nodes on the left hand side of the diagram, partitioned into ...
0
votes
1answer
435 views

Maximizing a function while minimizing one part of the same function

I have a function with two variables say $f(x,y)=f_1(x)-f_2(x,y)$ where $f_1(x)$ is the well known quadratic-form function in x while $f_2(x,y)$ is also a quadratic function in both x and y but not ...
0
votes
1answer
21 views

Why can't I use sum of probabilities as my loss function for machine learning?

I'd like to understand what is the major reason that we are using loss function of the following form in machine learning (I know it is obtained by taking a logarithm of the likelihood of the ...
1
vote
1answer
19 views

Can Somebody Help Me Find A Certain Paper about Hybrid Proximal Extragradient method for Bregman Functions?

I have read these two papers by Svaiter and Solodov. The first one, published in 1999 (http://pages.cs.wisc.edu/~solodov/solsva99Teps.pdf) presents an error criterion for the hybrid proximal ...
2
votes
2answers
30 views

Optimisation of a juice box: finding the least possible surface area that can hold the most volume

I have an investigation which requires me to design the dimensions of a juice box (cuboid) which has the least possible surface area that can hold the most volume. I am not sure as to how I should ...
0
votes
0answers
7 views

Can an infeasible point be used to initialize an Active Set Method (optimization)

Consider an optimization problem with a quadratic objective function and linear inequality and equality constraints. Consider an Active Set Method for optimization. Say you do not know a feasible ...
2
votes
0answers
38 views

Isotonic regression like

I have 2 ordered sets $$X=\{X_1<\dots<X_n\}$$ and $$Y=\{Y_1<\dots<Y_m\}$$ with $X_1<Y_1$ and and $X_n<Y_m$. I wish to approximate an increasing continuous function $g$ by piecewise ...
1
vote
0answers
18 views

Disadvantages of particle swarm optimization method

I am using particle swarm optimization method. It has a lot of advantages, but I am looking for disadvantages of this method. Can you help me?
2
votes
1answer
46 views

How do I classify extrema found using Lagrange multipliers?

Ok so I have found a bunch of local extrema using the method of Lagrange multipliers. Now how do I classify them as minimum or maximum? I cant use the second derivative test because its not a ...
1
vote
2answers
596 views

Absolute extrema

Find the absolute extrema of the function $f(x,y)=2xy-x-y$ over the region of the xy-plane bounded by the parabola $y=x^2$ and the line $y=4$ I was wondering if I needed to use Lagrange multipliers to ...
0
votes
0answers
6 views

Going from discrete solution to continuous solution (Dynamic Programming/Optimal Control)?

Suppose I have a discrete solution for a dynamic programming problem and an optimal control policy. If I can make the control policy continuous, by taking the limit as t-> 0, is that control policy ...
0
votes
1answer
20 views

Choosing the knots for a linear interpolation

I want to approximate a function through piecewise linear interpolation and try to understand how I could set the associated interval points optimally. Take a continuous function $f: X \rightarrow ...
0
votes
0answers
12 views

Exact and Heuristic Optimization Methods

Could anyone give me a rough classification for which kind of nonlinear- problems can I apply exact optimization methods (such as barrier function) and for which problems heuristic methods (such as ...
-1
votes
1answer
21 views

minimize the perimeter

Consider a window the shape of which is a rectangle of height $h$ surmounted by a triangle having a height $T$ that is $0.5$ times the width $w$ of the rectangle (as shown in the figure below). If ...
0
votes
1answer
700 views

How to maximize the volume of a cylinder with no top

A cylindrical can without a top is made using $A \text{ cm}^2$ of material. Find the dimensions that will maximize the volume of the can. What I have done was similar to the question: Optimization ...
0
votes
0answers
16 views

Minimizing sum of minimum

What are some theory/algorithm that talk about minimizing sums of minimums? For example, assuming y and z are discrete and the function is linear in x: $\min_{x} \sum_{y} \min_{z} f(x,y,z)$. I ...
0
votes
1answer
16 views

Normalizing eigenvectors when diagonalizing

Suppose your square matrix is symmetric and I want to diagonalize it. Why is it that at the end you normalize the eigenvectors to get your orthogonal matrix (actually orthonormal matrix)? Is this just ...
1
vote
1answer
20 views

Maximize sum of squares

Lets say that I know that $n$ values $x_i$ sums up to $\mu$: $$ \mu=\sum_{i=1}^n x_i $$ I also now that $0\leq x_i\leq 1$ for all $i=1\cdots n$. I want to find an upper bound as tight as possible ...
3
votes
0answers
162 views

How can I find a maximal inscribed ellipsoid to a *concave* set of points, in 3D?

I have a set of points which describe the surface of an irregular, natural (i.e., occurs in nature) object. This point set is not necessarily convex, and contains occasional indentations so parts of ...
0
votes
0answers
14 views

Optimization PDE

I have an assignment where the question reads: $\min J(u) = 1/2 \int_0^1x^2u'(x)^2 dx - \int_0^1 u(x) dx$ with $u \in H_0^1(0,1)$ show $J(u) \geq -1/2 \forall u \in H_0^1$ So I try the usual ...
0
votes
1answer
25 views

Show convexity of $f(x,y,z)= x^2+y^2+z^2+xyz$

Let $f(x,y,z)= x^2+y^2+z^2+xyz$. Show that $f$ is convex on $\Omega=${$(x,y,z)\in R^3 : x^2+y^2+z^2<\frac{5}{2}$}. To prove it, I want to show that $\nabla^2f(x,y,z)$ is positive definite. I ...
0
votes
2answers
2k views

Finding the center of mass of a cylinder

Help finding center of mass of soda can? If you represent the soda can as a right-circular cylinder radius=$4$ cm height =$12$ cm We are told to neglect the mass of the can itself. When the can ...
0
votes
0answers
16 views

Does the existence of a Algebraic Riccati Equation implies the existence of an functional minimization?

Let $\forall k\ge 0. V_k(x)$ be the value function related to the recursive optimization problem $ J(x_0) = \underset{u}{\inf} \sum_{k=0}^{N-1} x_k^T Q x_k + u_k^T R u_k + x_{N}^T P_N x_N \\ s.t. ...
2
votes
1answer
103 views

Solution for $\min_{x^Tx=1} x^TAx-c^Tx$

$\min_{x^Tx=1} x^TAx-c^Tx$ looks like a simple QPQC problem. If $A$ is positive semi-definite, can I get the solution by first getting $x=A^{-1}c$ and then projecting $x:=\dfrac{x}{||x||}$ to make ...
3
votes
1answer
410 views

Find maximum of a double integral over a region

I have a region given by $$R = |{ax}|+|{by}| \le 1$$ and $$f(x,y) = \iint\limits_{R}{(ax-by)^2 \ \cdot \ (3ab^3+12a^3b-6a^3b^2) \ \cdot \ \sin^2({\pi ax + \pi by}})dxdy$$ I need to find the values of ...
0
votes
1answer
40 views

How to rewrite/solve this differential equation

\begin{equation} \sin(\theta + d\theta) = \sqrt{1 + \frac{dy}{y}}\cdot{\sin(\theta)} \end{equation} I think this is a non-linear and non homogeneous first order equation. I found this whilst trying to ...
0
votes
1answer
23 views

Quadratic approximation of function of two variables near critical point

I know that to second-order, the Taylor polynomial of a function $f(x,y)$ at a critical point $(a,b)$ which gives a negative minimum is $$f(x,y)\sim f(a,b)+c_1(x-a)^2+c_2(y-b)^2+c_3(x-a)(y-b)$$ for ...
0
votes
0answers
40 views

Find a solution of optimal problem with an inequality constraint

Let $a,b,x$ be vectors in $R^n$, A be a matrix, $c,d \in R, c<d$. Solve the following problem: $$\begin{cases} \text{minimize} \quad (b-Ax)^T(b-Ax)\\ (a^Tx-c).(a^Tx-d) \leq 0 \end{cases}$$ Assume ...
2
votes
1answer
37 views

How many stationary points for this class of functions?

Let $f,g \in C^{\infty}[a,b]$ such that $f(a) = g(a)$ and $f(b) = g(b)$ and $f',g' \leq 0$ and $f'' > 0$ and $g'' < 0$. By the Rolle theorem I can say that there's at least one stationary point ...
-1
votes
0answers
45 views

Nonlinear constraints replaced by parameters and estimated iteratively

I have an optimization problem with nonlinear constraints in the following form: $x + y + 0.5(x+y)^2-z = 0$ $s+(x+y)*t\ge M$ I linearize these constraint by replacing the nonlinear terms by ...
0
votes
2answers
20 views

A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ true OR false

State true or false A differentiable function $f$ with maximum at $x=c\Rightarrow f''(c)<0$ I think this is a true statement but my book says this is a false statement.I do not understand why ...
1
vote
1answer
48 views

Ratio of two submodular functions is submodular?

Say we had 3 submodular functions $f(X)$, $g(X)$ and $h(X)$ is $\frac{f(X)}{g(X). h(X)}$ submodular as well? What can be said about the submodularity of $\frac{f(X)}{g(X)}$ and $f(X).g(X)$? I ...
0
votes
1answer
559 views

Maximize profit with dynamic programming

I have 3 tables… $$\begin{array}{rrr} \text{quantity} & \text{expense} & \text{profit}\\ \hline 0 & 0 & 0 \\ 1 & 100 & 200 \\ 2 & 200 & 450 \\ 3 & 300 & 700 ...
3
votes
0answers
57 views

Largest empty circle/sphere with non-polygonal location constraints

Finding Largest Empty Circles with Location Constraints contains the following theorem relating to the Largest Empty Sphere problem in 2-dimensions: Given a set S of n points and a k-gon P, the ...
0
votes
1answer
32 views

Linearize non-linear constraint [closed]

I have a problem which may be defined as: $$\max 5 x_{11} + 6 x_{12} + 2 x_{21} + 3 x_{22} \\ x_{ij}\in \{0,1\} \\ x_{11} + x_{12} = 1 \\ x_{21} + x_{22} = 1 \\ t_1,t_2 \text { integer} \\ (t_1 - ...
0
votes
1answer
21 views

Minimization problem with infinite variables and linear constraints

How can this minimization problem be solved? $$ \left\{\begin{matrix} \begin {aligned} &\sum_{i=1}^{\infty}P_i^3 \rightarrow min \\&\sum_{i=1}^{\infty}P_i=1 \\ &P_i\geqslant 0 \:for\: ...