Tagged Questions

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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9 views

Help in maximum likelihood estimation of distance

The model generating the observation is of the form $y_n = A^Tx_n + U_n$ where $x$ is the output of a a linear stationary model and $U$ is a zero mean Gaussian noise of known variance. The set of ...
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0answers
13 views

Variation of Bin-packing with classes of bins and objects

I'm working on a problem that is a variation of bin-packing, but a bit more general form with extra constraints. The problem definition is as follows- We have objects of varying sizes, which can be ...
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2answers
18 views

Intersection of two lines and the minimum of the sum of the two.

We use a formula in my Operations Research class for finding the 'Economic Order Quantity', given the cost function (sum of Holding and Ordering costs) $$C = \frac{Q}{2}H+\frac{D}{Q}S$$ where $Q$ is ...
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1answer
18 views

Maximize parial sum of binomials

During my research I got to the point when I need to find $$ \arg \max_w \left( (n-w) \sum_{j=0}^d \binom{w}{j} \binom{2^r - (j+1) 2^{r-j-1}-2}{t} \right) $$ with respect to $w$ only (i.e. $d$, $n$, ...
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0answers
11 views

constraint optimization: sparsity with non zero constraints

I have an obtimization problem in the following form. $\min f(x)\\ s.t \|x_i\|_0\leq\lambda\\ x_i \geq0\\ \sum_i x_i = 1$ where $f(x)$ is convex. What is the easy way to optimize it as I have a ...
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15 views

Affine functions as equality constraints in convex optimization problems

I am studying on an introduction to convex optimization problems. When defining a convex optimization problem, we have a convex object function, $f(x)$, a set of convex functions $g_i(x)$ where the ...
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0answers
20 views

Is there a nice representation for KKT conditions for matrix constraints?

I have a convex programming problem: $\min \left\lVert J - R \right\rVert _2$ $J,R$ are matrices. $J$ is given for the problem. One of the constraints is: $R = KQ$ Here, $R,K,Q$ are matrices. $K$ ...
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20 views

What is the constraint matrix in the assignment problem formulation?

The linear assignment problem may be posed as $$ \min_{X\in \mathbb{R}^{n \times n}} <X, C>_F, \;\; \text{subject to}\;\; \sum_jx_{ij}=1, \forall i \; \sum_i x_{ij}=1, \forall j \;\; x_{ij} ...
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1answer
24 views

Proof concerning basic solutions

Prove that every basic solution of $Ax=b$ (where $A$ is a matrix of rank $r$) is set by $r$ linearly independent columns of matrix $A$ (so it is $[A^{k_1}\dots A^{k_r}]\bar{x}=b$ where $A^{k_1},\dots ...
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0answers
20 views

Find non degenerate linear programming problems

I have to find non degenerate linear programming problem in a canonical form such that: a) it has no solutions b) it has solutions, but but doesn't have an optimal solution A ...
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1answer
14 views

Find global and local maxima and minima, given the graph of the function

My attempts were local max: 3,8 - local min: 5 - global max= 3, global min= 5 Module is giving me incorrect. No partial credit. So I can't tell where the problem lies. local max: 6, 4.?? local ...
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18 views

What dosage maximizes sensitivity?

The sensitivity of the body to the drug is defined as $dT/dD$. For some positive constant $C$, a patient’s temperature change, $T$ , due to a dose, $D$, of a drug is given by $$T = \left(\frac C2 − ...
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1answer
21 views

L(x)<=U(x) & L'(x) and U'(x) exist. Assume there's a pt c st. U(c)=L(c). Consider U(x)-L(x) and show that c is a min of this function

Question and attempt at question are in the photo below. I have gotten halfway through but I am confused how to show the rest of the question (mainly part c) Thanks for your help in advance :)
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1answer
23 views

How to find this maximum?

How to find the maximum of $$f = |x_1x_3 + x_1x_4 + x_2x_3 -x_2x_4|$$ on the four-dimensional cube $\{x \in \mathbb{R}^4:|x_k| \le 1, 1 \le k \le 4\}$? Calculations with CASes suggest it equals 2.
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1answer
19 views

Prove that the solution of a system is the same as minimum of the sum of squares

I was reading through some results on google to learn simple optimization problems in MATLAB, and found a PDF where a method is described which I don't understand. Here is a picture of it, or see PDF ...
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1answer
33 views

maximization using Lagrange

I am maximizing $f(x,y)=-x$ given the constraint $g(x,y)=x^2-y^2=0$ To satisfy the non degenerate constraint qualification I have: $Dg(x,y)= [2x\quad-2y]$ and the set of $(x,y)$ that satisfy it ...
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0answers
16 views

Master of Science in Operations Research [on hold]

Does any one have any ideas in regards to doing a Msc thesis in Operations Reserach field area analytic Hierarchy Process?
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4answers
35 views

Finding maximum area of rectangle with constraint

Find the maximum area of a rectangle in the xy-plane with its sides parallel to the axes, one vertex at the origin, and the diagonally opposite vertex on the curve $$ x^2 + y = 1 $$ I am supposed ...
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0answers
11 views

Find maximum hyperplane separating two classes by optimizing the Langrangian function.

I am trying to solve the following problem: I am having difficulty starting. I know this is a constrained optimization problem for support vector machines. I am wondering how the training data is ...
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1answer
22 views

Simplex method and basic solutions

I have put this into the form $0.5x_1 + 0.25x_2 + x_3=6$ $-x_1 - 3x_2 + x_4=-2$ $x_1 + x_2 = 10$ Is this correct? If so, how do I find a basic solution so that I can begin the simplex algorithm? ...
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13 views

$f(x)\leq c_1\max(e^{c_2\cdot x^2},e^{-c_2\cdot s \cdot x})$

Let $f:(0,\infty)\rightarrow \mathbb{R}$ be such that $$f(x)\leq e^{C \cdot t^2\cdot s^2-t\cdot x\cdot s},$$for all $t\in[0,1]$ and for some positive $s,C$. I have to prove the bound $$f(x)\leq ...
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0answers
17 views

Noisy pencil of planes

I have a system of $N$ equations of planes in the form: $\begin{pmatrix} a_1 & b_1 & c_1\\ \vdots & \vdots & \vdots \\ a_N & b_N & c_N\\ \end{pmatrix} \begin{pmatrix} x\\y\\z ...
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0answers
17 views

Reduce degree of a high degree unconstrained binary term to quadratic unconstrained binary term

I'm working on a optimization project, in this project I have to convert higher order unconstrained binary polynomial to quadratic unconstrained binary polynomial. Can anyone give me a hint of how to ...
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1answer
52 views

Converting a Summation to an integral

Please how do I convert this summation $$ \frac{r-1}{n} \sum_{i=r}^n \frac{1}{i-1} $$ to the integral $$ x \int_x^1 \frac{1}{t} dt = -x \ln x? $$ by substituting $x = r/n$, $t=1/n$ and $dt =1/n$. ...
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1answer
15 views

Given $r_{i}>0;\: i=1\ldots K$ when $\frac{\sum_{i=1}^{K}l_{i}}{max(\frac{l_{1}}{r_{1}},\ldots,\frac{l_{K}}{r_{k}})}$ is maximized?

Given $r_{i}>0;\: i=1\ldots K$ , We want to determine $l_{i}>0;\: i=1,\ldots,K$, such that function ...
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0answers
17 views

A sufficient condition for a good to be normal

Context: there are $2$ goods with prices $P_1$ and $P_2$ and the decision maker has the utility function $U(C_1,C_2)$. Denote $U_j=\frac{\partial U(C_1,C_2)}{\partial C_j}$ for $j\in\{1,2\}$. A good ...
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0answers
21 views

Basic linear optimisation problem

I want to solve the following problem: A company uses cement and three different ways of production $P_1$, $P_2$ and $P_3$ to produce the products P,Q,R and S. For one ton of cement, the ...
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0answers
11 views

maximization of a function with random variable

I would like to know whether this is true in general, and if not when this can be. I am not sure and so I am mostly asking for confirmation. So, is the following correct ? $$\log [\max_{x} ...
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0answers
23 views

Legendre transform and Minimax Theorems.

Denote the class of lower-semi-continuous convex functions $f:\mathbb{R}^n\to \mathbb{R}\cup\{\pm\infty\}$ by $Lscx(\mathbb{R}^n)$ ( so that only function attaining the value $-\infty$ is the constant ...
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1answer
61 views

How to find minimum?

Is it possible to find exactly (not numerically) the minimum of the function $$\sqrt{15-12\cos(x)}+\sqrt{4-2\sqrt{3}\sin(x)}+\sqrt{7-4\sqrt{3}\sin(x)}+\sqrt{10-4\sqrt{3}\sin(x)-6\cos(x)} $$ on the ...
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0answers
27 views

Unconstrained Optimal Control - $J = \frac{1}{2}x^2(2) + \frac{1}{2} \int_{0}^{2}(u^2 - 2xu)dt$

I've been given the following unconstrained optimal control problem, but I feel like I've made a mistake at some point. The system $\dot x = -x + u$, where u = u(t) is not subject to any ...
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1answer
30 views

Minimize distance to a given point subject to a number of linear inequality

I'm trying to find a point that has minimal distance to a known point and satisfies a number of linear inequality. Example in two dimensions and one inequity: $min\{$distance to $(50,70)$ | ...
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1answer
30 views

Deleting 0's from a random mod 2 matrix

I am fairly new to optimization problems, so please forgive my lack of knowledge. That said, I'm trying to write a program that takes an NxM matrix randomly filled with 0's and 1's, then reduces this ...
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0answers
47 views

Mixed-Integer Linear Programing : get the maximum constant associated to a non null variable

Does anyone know a way get the maximum constant associated to a non null variable using Mixed-Integer Linear Programing ? I would like to get the variable $a$ in this description : $$ i = 1,\ldots,m ...
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0answers
14 views

Can this specific Linear Program constraint be expressed? [duplicate]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
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1answer
15 views

What is the difference between a local maximum and an unconstrained local maximum?

I can see that the definition of local maximum and unconstrained local maximum is written differently, but to me they look like they are defining the same thing. Furthermore, based on Fig 4.1, it ...
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0answers
79 views

Can this specific Linear Programming constraint be expressed? [closed]

Thanks for your time. I have a linear program and no idea how I could express a form of constraint and even if it's possible. Maybe someone here know a solution. A company assembly and sells a ...
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1answer
20 views

What are the minimum and maximum of $x$ and $y$ within the set $ 0 \le x \le 2$, $x - 2 \le y \le x$?

Given a set, how do I calculate what it's minimum and maximum is for x and y? $$ 0 \le x \le 2 \ , \ x - 2 \le y \le x$$ I informally look at it and think "if x is 0, then y is at most 0, and least ...
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1answer
31 views

A question on a nonnegative quadratic form

Denote $x,y,z$ as variables, and $a,b,c$ as coefficients. Suppose $a\leq b\leq 0\leq c$ and $a+b+c=0$. Could anyone help me prove whether the following quadratic form positive semi-definite? ...
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2answers
29 views

maximum, complex quadratic function, Is my solutions correct?

I'm trying to compute $\max_{|z| \le 1} |(z+2)(z-1)|$. Here's how I do it: $\{z \in \mathbb{C} \ | \ |z| \le 1 \}$ is compact and $f(z) = (z+2)(z-1)$ is continuous, so it suffices to look for ...
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0answers
19 views

Gradient and hessian of square of quadratic form

I'm trying to differentiate a term of the form $(x^TA x)^2$ where $x$ is a vector and $A$ is a symmetric square matrix. Can anyone please tell me what the gradient and Hessian matrix of this term ...
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1answer
38 views

Optimization problem: rowing across a lake

A woman at a point A on the shore of a circular lake with radius $r=3$ wants to arrive at the point $C$ diametrically opposite $A$ on the other side of the lake in the shortest possible time. She ...
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1answer
31 views

Finding maximum of the basic Bernstein Polynomials

The basic Bernstein Polynomials $B_{n, k}$ are defined for all integers $n, k$ with $0 \leq k \leq n$ by $B_{n, k} = {n \choose k} x^k (1 - x)^{n-k}$ for $x \in [0,1]$. I want to prove that the ...
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1answer
26 views

Newtons method for optimization

How can I solve this question? Use Newton's method for a system to write $x^2+y^2=25$ and $x^2-y=2$ in the form $J*\delta=-f$. Define the matrix $J$ and vectors delta, $f$. Dont perform iterations.
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0answers
63 views

Shortest distance from a point to vertices of a cube

A $d$ dimension cube has vertices $P_1,...,P_{2^d}$, where the coordinates of each vertex are either $0$ or $1$. To find which vertex of $P_1,...P_{2^d}$ is closest to a given point $P=(p_1, ...
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0answers
23 views

A question about the definition of Lipschitz continuity

Suppose $f(x,y)$ is a function on $R^2$. If $f$ is Lipschitz continuous with respect to $y$, then $|f(x,y_1)-f(x,y_2)|<C|y_1-y_2|$ for some constant $C$. But can anyone tell me whether the ...
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1answer
31 views

What optimization problem is this?

Minimize $$\sum_{i=1}^{m}w_i x_i$$ with $w_i \in \mathbb{Z}_{\ge0}$, and $x_i \in \{0, 1\}$ subject to a set of $n$ conditions of the form $$\sum_{i\in S_k} x_i \equiv c_k \pmod{2}$$ for $S_k ...
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0answers
11 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
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1answer
35 views

What is the most efficient way of determining a date of birth using yes/no questions?

First question here on StackExchange! Sorry if this question is not quite of the correct style - please let me know if so. Anyway, here's the context. I'm trying to write a program to determine a ...
0
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1answer
13 views

Relation between minimum of a function and minimum of the sum of the same function and a linear term

I'd like to know if it's true that if given a function $f(x):X \mapsto \mathbb{R}$ and a vector $c \in X$, then if $$v = \arg\min_x f(x) + x^tc$$ one can say that $$v-c = \arg\min_x f(x)$$ Does this ...