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
1answer
92 views

Why is this a quadratic programming problem?

I am sorry if this is a stupid question, I'm very new. How would I minimize the following objective? $\sum_{k=1}^p\| I_{k} - M_{k}A \|^2$ Each I and M are known. I am told I can use a quadratic ...
1
vote
0answers
102 views

Algebraic manipulation of Lyapunov function

I have a problem I would like some feedback on. I have spent 6 hours on it examining various techniques (numerically and analytically). I need to find the values of $k$ for which $x^2+ky^2$ is a ...
2
votes
1answer
148 views

Set convergence and lim inf and lim sup

I'm a bit confused with the general concept of convergence of a sequence of sets. I'm well aware that the limit of a sequence $\{C^{\nu}\}$ exists iff $$\liminf_{\nu \rightarrow \infty} C^{\nu} = ...
2
votes
1answer
202 views

Optimisation Problem on Cone

The problem I've got here is to prove that semi vertical angle of a cone with maximum volume with total surface area constant is equal to $arcsin(\frac{1}{3})$ I am trying to do that by making some ...
0
votes
0answers
22 views

Trust region sub-problem Explicit Formula [duplicate]

Consider the $2 \times 2$ trust region sub-problem. Given $Q$ symmetric $2 \times 2$, vector $\mathbf b$ and $\Delta > 0$, find $\mathbf x$ that minimizes $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x ...
0
votes
1answer
71 views

Trust region sub-problem with Jacobi Condition

Consider the $2 \times 2$ trust region sub-problem. Given $Q$ symmetric $2 \times 2$, vector $\mathbf b$ and $\Delta > 0$, find $\mathbf x$ that minimizes $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x ...
3
votes
2answers
65 views

Matrix which when multiplied, gives a maximal minimum of elements of result.

I'm working on an optimization problem and am stuck at this particular step. Let $\bf{A}$ be a matrix with 4 columns and a finite number of rows, consisting of elements which are either 0 or 1. Let ...
3
votes
1answer
58 views

Nonconvex set converging to a convex set despite holes

I'm looking at the example in Figure 4-7 of "Variational Analysis" (Rockafellar and Wets). Basically, there's a sequence of sets $C_{\nu}$ riddled with holes, and it states that the sequence ...
1
vote
1answer
19 views

How to recognize if an algorithm working on ordinal data will also work if the ordering is reversed?

Inspired by a comment on this question. Assume that I have an algorithm which uses ordinally scaled data. The algorithm in the original question was the solution of the Secretary Problem. It uses ...
1
vote
0answers
52 views

Sequence of solutions of approximated problem converging to stationary solution of the original problem

Let $I(x)$ be the indicator function defined as $$I(x) \triangleq \begin{cases} 1, & x \geq 0 \\ 0, & x < 0 \end{cases}$$ and let $L(x,\alpha) \in [0,1]$ be a smoothing function that ...
0
votes
1answer
43 views

Sequence of connected sets whose limit is disconnected

Let us define $\mathcal{X} = \{0,1\}$. Is it possible to define a sequence of connected sets $\{ \mathcal{X}_{\nu} \}$, with $\mathcal{X}_{\nu} \subseteq [0,1]$, such that $$\lim_{\nu \rightarrow + ...
3
votes
1answer
93 views

Minimizing distance of circles from points without overlapping

I am designing a user interface, and I have encountered the following problem: I have $p_1 ... p_n$ points in $\mathbb{R}^2$, and $c_1 ... c_n$ circles with constant $r$ radius. I want to minimize ...
0
votes
2answers
378 views

Lagrange Multipliers to find shortest distance

Use Lagrange multipliers to determine the shortest distance from a point $\,x \in R^n\,$ to a plane $\{y\mid b^Ty = c\}.$ I don't even know where to start!
3
votes
1answer
376 views

Absolute values in linear programming

Suppose I have an objective function in my LP as follows $max$ $|x|$ Based on some googling, I have found there are two ways to convert this into a standard LP. Method 1. $|x|$ = $ x^+ + x^-$ $x ...
0
votes
1answer
102 views

Explain the convexity by looking the hessian matrix of a function

The hessian matrix of a function is given by, $$ H = \begin{bmatrix} a & b & c \\[0.3em] b & b & 0\\[0.3em] c & 0 & c \end{bmatrix} $$ where, ...
6
votes
2answers
164 views

Estimating the maximum of a Brownian motion over the unit interval

Let $\left(B_t\right)_{t \in \left[0,\infty\right)}$ be a standard Brownian motion over the probability space $\left(\Omega, \mathcal{A}, P\right)$. For each $x \in \left(0, \infty\right)$, give an ...
6
votes
2answers
379 views

Intuition on why the average minimizes the euclidean distance

I saw that there was a question with a very similar (if not identical) flavour to my question, but the answer was derived from derivative, the method that I already knew solved this problem. I feel ...
1
vote
1answer
566 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
0
votes
2answers
2k views

Finding the Absolute Maximum and Minimum of a 3D Function

Find the absolute maximum and minimum values of the function: $$f(x,y)=2x^3+2xy^2-x-y^2$$ on the unit disk $D=\{(x,y):x^2+y^2\leq 1\}$.
3
votes
0answers
89 views

Convergence of sequences of sets

Assume we have the indicator function $$I(x) \triangleq \begin{cases} 1, & \quad x \geq 0 \\ 0, & \quad x < 0 \end{cases}$$ and the sequence of its approximation functions $\{ I_{\nu}(x) ...
2
votes
1answer
81 views

Combination Problem with mulitiple variables

I am new to this, but getting into math more and have a question regarding combinations and permutations with several variables involved. I work for a sales company and this question is based on ...
2
votes
0answers
66 views

Sequence of connected sets converging to disconnected set

Let us have the disconnected set $\mathcal{X} = \{0\} \cup [\underline{x},\overline{x}]$ and its approximation $\mathcal{X}_{\nu} \subseteq [0,\overline{x}]$, with $0 > \underline{x} > ...
2
votes
1answer
109 views

Maximum and minimum of weighted sum

For $w_i\ge 0$ and some constants $\alpha_i , i=1,...,n$, what is the maximum and minimum of $\sum_{i=1}^{n}\alpha_i w_i$ subjected to $\sum_{i=1}^{n}w_i=1$? Intuitively, I put all weight on the ...
4
votes
1answer
103 views

Is this Frechet derivative correct?

Problem statement: Let $u \in L^2[0, 1]$ and $$J(u) = \int_0^1 u(t) u(1-t)dt$$ Find $J'(u)$ and $J''(u)$. Attempted solution: First derivative There is a hint that the derivative looks like this: ...
0
votes
1answer
93 views

Stock cutting and column generation giving suboptimal answers?

I'm doing a stock cutting implementation. I use the delayed column generation approach. I'm getting suboptimal answers with the following simple case: raws length: 630 in. demands: 10 x ...
0
votes
1answer
80 views

How to solve this optimization problem?

Suppose I have the following problem: Maximize: $\quad\quad x_1+x_2+x_3+x_4$ Subject to: $\quad\quad \dfrac{\gamma\;a_1\;x_1}{\gamma\;a_2\;x_4+1}\geq1$, $\quad\quad\quad\quad\;\;\quad\quad ...
0
votes
1answer
62 views

A Constrained gradient descent Algorithm

I'am looking for a way to find a solution to the constrained minimization problem using the gradient descent Algorithm. it follows ...
1
vote
1answer
195 views

How do I setup the lagrangian for this problem?

I have a function $y(x)$, that I would like to maximize, subject to two constraints. It is given by: $$ \max_{x} \ y(x) = a \ cos(x) + b \ sin(x) \\ \text{subject to:} \\ x \geq 0 \\ x \leq ...
3
votes
2answers
184 views

significance of zeros of a transfer function?

In control theory, the poles of a transfer function give information about the stability and behavior of a system. I'm not sure and can't find anywhere what the significance of the zeros of a ...
0
votes
1answer
136 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
0
votes
1answer
96 views

Minimize a trig function. Getting stuck.

So I have just about given up on this. Here is the problem. FYI, all angles are in degrees, and $L$, $R$ are just strictly positive scalars. I have a trig-function $D$. Its derivative shown below, ...
0
votes
1answer
54 views

Optimization using Lagrange Multipliers for conditions with different codomain

I'm trying to maximize the trace of $X^TAX$ subject to the columns of $X$ being orthonormal, where $A$ is a diagonal matrix and X is not necessarily square, but does not have more columns than rows. ...
1
vote
1answer
52 views

The most optimal way to solve this set of non-linear equations in high dimensions

So I have a series of non-linear equations which I wish to solve as fast as possible, to illustrate for the case of $n = 4$, I have the following equations: \begin{gather*} ...
1
vote
1answer
23 views

Null space and minimization

Let $x^*\in\text{argmin }f(x)=\text{argmin }\frac{1}{2}\|Ax-b\|^2$ where $A$ is a linear operator. Show that $\text{argmin }f=x^*+\text{Null}(A)$. For $x\in x^*+\text{Null}(A)$ we have ...
0
votes
1answer
92 views

A simple optimization problem

$$f = x^Tx$$ $$g = Ax-b $$ The constraint is $Ax-b = 0$ I calculated $J' = f'+\lambda g'$ which is $2x^T+\lambda A^T = 0 $ and $Ax-b=0$ . I dont know what to do next please help me out .
0
votes
2answers
49 views

Lagrangian with inequalities

I have a toy question on SVM , where i have to find the weight $w$ by solving the Lagrangian multiplier method by hand . I know Lagrangain with equalities only . Here I have to deal with inequalities ...
-1
votes
1answer
45 views

value of x when equation reach maximum?

how can i find the value for $x$ when $y$ reach to $maximum$ value or go to $infinity$ $$ y=\frac{1}{2}A(1+\cosh{(x \sqrt[2]{\frac{B}{A}})} )$$ where $A,B$ are constant $x$ range from $0$ to ...
0
votes
1answer
49 views

How large is the largest prime required to satisfy these requirements?

I require a set of primes, all being equal to or greater than $2v+2$. The product of the primes should be at least $(2^v)+1$. I have one additional constraint. Each prime minus one must be ...
0
votes
1answer
482 views

Minimized sum of the distances with street distance

This exercise comes from Bazaraa Linear Programming and Network Flows book : Consider the problem of locating a new machine to an existing layout consisting of four machines. These machines are ...
2
votes
3answers
372 views

Max perimeter of triangle inscribed in a circle

What is the maximum perimeter of a triangle inscibed in a circle of radius $1$? I can't seem to find a proper equation to calculate the derivative.
3
votes
2answers
69 views

Finding minimum $\alpha > 0$ so that $\det(A - \alpha B) = 0$ for positive definite $A,B$

Given two positive definite symmetric matrices $A,B$, I'd like to find the minimum $\alpha > 0$ such that $A - \alpha B$ is singular, i.e., the threshold where $A - \alpha B$ is no longer positive ...
0
votes
1answer
149 views

Optimization of difference of two convex function.

Suppose we have to minimize a function $f(X,Y)-g(X,Y)$, where both the functions are convex. How can this be solved in matlab? Is there any tool to solve this in matlab?
1
vote
0answers
63 views

Multivariate Dynamic Optimization Model with Costs

I'm looking for kinds of models able to describe a situation such as the one described below (what I'm actually dealing with is load balancing but this is a more concrete example): We've a got a ...
4
votes
1answer
190 views

Minimize Frobenius norm with unitary constraint

I am trying to find a unitary tramsformation, $M$, that minimizes $\Vert MA-B \Vert_F^2$ where $A$ and $B$ are $N\times L,\;L\ge N$. I know how to solve it without the unitary constraint. I thought ...
1
vote
1answer
50 views

Minimizing difference and individual variables in convex problem

Let's say I have the following optimization problem: $$ \begin{align*} \min_{\mathbf{x},\mathbf{y}} & \sum_i x_i-y_i \\ \mathrm{s.t.} & \{\mathbf{x},\mathbf{y}\} \in ...
0
votes
2answers
125 views

Symmetric Positive Definite and Gradient Proof

I have the function $f(x)=\frac {1}{2} \mathbf x^T Q \mathbf x - \mathbf b^T \mathbf x$ where $Q$ is symmetric. I'm trying to show that solving $\nabla f(\mathbf x) = 0$ is equivalent to solving $Q ...
1
vote
1answer
217 views

Solution to a Hadamard product least squares

Given two full-rank matrices $A \in \mathbb{R}^{n \times p}, B \in \mathbb{R}^{n \times k}$ and vectors $y \in \mathbb{R}^n, u \in \mathbb{R}^p, v \in \mathbb{R}^k$ I'd like to solve an optimization ...
0
votes
1answer
117 views

Verify by Second Derivative Test

$$A(x)=2\sqrt{x^2-16}+\frac14\sqrt{68x^2-x^4-256}\;,\;\; (4 < x < 8)$$ of which the derivative is: $$a'(x)=\frac{2x}{\sqrt{x^2-16}}+\frac{136x-4x^3}{8\sqrt{68x^2-x^4-256}}$$ I first had to ...
1
vote
1answer
27 views

Proxy optimisation problem

Suppose we have a set of participants $p$ who should attend $e$ number of events and everyone of them must declare his presence with signature. Each can however sign for $s$ number of other ...
0
votes
1answer
111 views

Relax equality into inequality in convex problem

Let $\mathbf{x}, \mathbf{z}, \underline{\mathbf{x}}, \overline{\mathbf{x}} \in \mathbb{R}^{I}$, where the first two are variables and the last two are given data. I have the following problem: ...