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
5 views

How to efficiently compute the pareto front in a >2 dimensional multi-objective case?

I'm currently working on an optimization problem with 4 different objective functions and need an algorithm to compute the pareto frontier from several "solutions" to that problem. I already found ...
1
vote
0answers
22 views

Optimization problem, how to obtain the optimal F?

The objective is as follows: $\min_{\mathbf{F}} a Tr(\mathbf{F} \mathbf{F}^H) - \mathbf{b}\mathbf{F}^H \mathbf{C} \mathbf{F} \mathbf{d}$ $s.t.\ \ \ Tr(\mathbf{F} \mathbf{F}^H)<p$ where $a$ and ...
1
vote
0answers
19 views

Can difference of the $log$ function be approximated?

I am currently trying to optimize a problem. $$\text{ArgMax}_x \log(1+f_1(x))-\log(1+f_2(x))$$ Due to the fact that $\log (x)$ is a monotonic increasing function, this is equivalent as to ...
0
votes
0answers
47 views
+100

How exactly do I prove that I find the maximum of the function

I am currently trying to maximize an objective function $f(a,b,c,d,e)$ over the variable $b$ only. By taking the derviative of f over b, setting it to zero, I can solve b in terms of the other 4 ...
1
vote
1answer
56 views

Maximum determinant of a $m\times m$ - matrix with entries $1..n$

I want to find the maximal possible determinant of a $ m\times m$ - matrix A with entries $1..n$. Conjecture 1 : The maximum possible determinant can be achieved by a matrix only ...
0
votes
1answer
28 views

Constraineed Maximization Problem

I am creating a game, and have run into quite a tricky problem which I have been wrestling for days. I have been able to turn into somewhat mathematical terms (bare with me, I'm a programmer not ...
2
votes
1answer
295 views

Maximum likelihood estimators, hypergeometric and binomial

I'm trying to solve a two part problem. The set up is as follows: consider a bag with $\theta$ red marbles and $7-\theta$ blue marbles, with $\theta$ being unknown. Let $x$ denote the number of red ...
3
votes
3answers
123 views

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac n2 \rceil$ or $ \lfloor \frac n2\rfloor $?

How do you prove ${n \choose k}$ is maximum when k is $ \lceil \frac{n}{2} \rceil $ or $ \lfloor \frac{n}{2} \rfloor$ ? This link provides a proof of sorts but it is not satisfying. From what I ...
1
vote
3answers
36 views

Greatest value of the binomial coefficient. [duplicate]

How should I prove the greatest value of the binomial coefficient $C(n,r)$ occurs for $r=\left[\cfrac{(n+1)}{2}\right]$ ?
0
votes
1answer
32 views

Global/local optima for this function

I have the following function $f(x_1,x_2) = \frac{x_1}{x_2+p} + \frac{x_2}{x_1+p}$ where $x_1$ and $x_2$ $\in$ $[0,1]$ and $p > 0$ is a constant I want to find global/local maxima for this. ...
0
votes
0answers
13 views

Upper bound on optimal multinomial logit

Let $[N]={1,...,N}$ denote a set of items, item $i$ has a unit revenue of $r_i>0$ and a utility $u_i>0$. Items have to be assorted in $N$ slots with sampling probabilities $v_k>0$. Let ...
1
vote
1answer
15 views

Method to calculate the best way to repay two different loans given a set amount of money per month?

Given two (or more) loans of different balances and interest rates and a single amount of funds available per payment period, is there a way to calculate the best way to split the available funds to ...
0
votes
1answer
23 views

Minimizing sum of functions implies minimizing their squares, maximizing the sum of the inverses?

I have $n$ functions (Say $f_1\space to \space f_n$) of $k$ variables (Say $x_1\space to\space x_k$) each. The functions are all positive, as well as the variables $xi's$. I do not have explicit ...
1
vote
1answer
27 views

Minimization with two functions that are not completely related

Two caveats: 1) This is a problem I formulated myself, and so may not be structured correctly/logically. 2) I don't have an extensive math background, but am currently finishing up Calc 3. I have an ...
2
votes
1answer
23 views

Horn–Schunck method. Explanation of iterative solution

I am reading this paper (explanation of Horn-Shunck method for finding optical flow) and trying to understand it. My stumbling block is obtainig solution of system of linear equations I(x, y, t) ...
2
votes
1answer
20 views

normal equations of $ y(t) = \gamma e^{\lambda t} $ for minimizing the error

Let $ y(t) = \gamma e^{\lambda t} $ and we have the points $(0,2)\ (1,0.7)\ (3, 0.3)$. The task is to get the parameter so that error is minimal. So we need to get the matrix for the normal ...
0
votes
2answers
54 views

Finding $a$ and $b$ knowing the maximum [on hold]

The function $f(x)$ = $ax e^{bx^2}$ has the maximum value $f(2) = 1$ where $a$ and $b$ are real numbers. Find $a,b$.
0
votes
1answer
687 views

How to describe minimization of L1 norm error using linear programming?

Given a set of $n$ pair points $(x_1, y_1), ..., (x_n, y_n)$ in the plane, I need to find a line $ax + by = c$ that fits the points of the L1 norm error points as closely as possible. I need a linear ...
-1
votes
3answers
29 views

Optimization with contraint

Given the value K with constraint x+y = K, what can be the maximum value of x*y be? How did they derive this answer? It is equivalent to finding the maximum value of x*(K-x), which will happen when x ...
1
vote
0answers
52 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
1answer
48 views

Proving boundedness of a function (part 1).

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...
0
votes
0answers
38 views

Strategies to work with system of trigonometric inequality

I'm trying solve this problem using matlab, anybody know good strategies to work with system of trigonometric inequalities such as $ ...
0
votes
0answers
49 views

Find scaling factor that minimizes f(x) - round(f(x))?

Let's say I have a function $f(x)$, which has a fractional component $\{ f(x) \} = f(x) - \lfloor f(x) \rfloor$. I would like to add a scaling factor $h(x)$, where $h(x)$ is a polynomial, such that ...
1
vote
2answers
84 views

Unsolvable(?) Assignment Problem

I've recently been trying to implement the Hungarian Method in C++, and I've been using 5x5 matrices to test my program. Last night I came across a matrix which neither I nor my program can solve. Is ...
0
votes
0answers
27 views

sum of logarithms of linear-fractional functions Optimization Problem

I am new to optimization theory and I am facing this optimization problem. \begin{equation} maximize \qquad f(x) = \sum_{i} ...
0
votes
0answers
26 views

multivarable optimization problem, what is the procedure?

Sorry for this obvious question. I am trying to maximize an objective function that consist of 5 variables (a,b,c,d,e) over a and b. That is , $max _{a,b}f(a,b,c,d,e).$ So I procedure I took is ...
0
votes
0answers
28 views

compute a certain maximum in MATLAB

let $ C \in \mathbb{N} $ and $ c_1>c_2>\ldots>c_k \in \mathbb{N} $ with $ C>c_1 $ and $ c=(c_1,c_2,\ldots,c_k)^\top \in \mathbb{N}^k $, where $ \mathbb{N} $ are the natural numbers without ...
0
votes
1answer
22 views

Volume of a polytope cut off by a hyperplane

Given a maximization problem with constraints, and adding a few more constraints using the Gomory cuts and solving the relaxed maximization problem, we can arrive at integer solutions. I am looking to ...
0
votes
1answer
20 views

Closest Positive-Definite Matrix Subject to a Contraint

Given a positive, semidefinite, real 2n by 2n matrix $A$, is there a formula or an algorithm that finds the closest (in some sense, preferably Frobenius distance) positive, semidefinite, real 2n by 2n ...
0
votes
0answers
26 views

Dimensional Consistency in Grids used in Optimization

I am working on an optimization problem in the research I am doing and my partner and I have found that in order to quickly converge on a solution using a specific PSO (the firefly algorithm - it's ...
3
votes
3answers
457 views

How to find the minimum value of the expression?

Let $x$, $y$, $z$ be three nonegative real numbers and $x^2 + y^2 + z^2 = 5.$ Find the minimum of the expression $$E=\dfrac{1}{2}(x^2 y^2 + y^2 z^2 + z^2 x^2) + \dfrac{96}{x + y + z + 1}.$$ What ...
1
vote
0answers
18 views

Find unknown such that four dependent quantities have the same value.

I have $12$ unknown $a_i, b_i, c_i, i=1,\ldots,4$, that should satisfy equations $$ \sum_{i=1}^4n_ia_i=a,\quad\sum_{i=1}^4n_ib_i=b,\quad\sum_{i=1}^4n_ic_i=c, $$ where $n_i,\,i=1,\ldots,4$ and $a,b,c$ ...
3
votes
0answers
31 views

Strong duality: When does the optimal primal variable coincide with the primal variable giving the dual function.

I'm considering the inequality-constrained optimization problem of finding $$ x^{\star} = \arg \min_{x} f(x) \;\; \text{s.t.} \;\; h(x) \le 0 $$ which is assumed to have a unique minimizer. The ...
0
votes
4answers
39 views

Given a satisfactory real number = [any integer]/(2b) where a and b are integers, how would one find the minimum value of b?

For instance, 0.625 = 5/(2*4). Given 0.625, how would one find 4? 0.75 = 1/(2*2). Given 0.75, how would one find 2? I should ...
4
votes
1answer
413 views

An optimization problem involving orthogonal matrices

Let $X\in\mathbb{R}^{3\times 3}$ be an orthogonal matrix. Then $\mathrm{vec}X\in\mathbb{R}^9$ is a 9 by 1 vector formed by stacking the columns of the matrix $X$ on top of one another. Given a matrix ...
0
votes
0answers
16 views

Hyper-plane that separating hyper-cube.

Suppose $\Omega \in \mathbb{R^4}$ is closed unit ball in $ ||.||_{inf}$ i.e. Hyper-cube. 1) Am I right that there are L=16 extreme points of $\Omega$, all are vertices of the hyper-cube. 2)Is it ...
1
vote
2answers
26 views

How to introduce flat cost of flow over a node using mixed integer programming.

In the set up for the program we have a graph where we are trying to minimize the cost of sending flow over the arcs. I have formulated the following linear program. \begin{array}{ll} \text{minimize} ...
1
vote
2answers
33 views

A curious question about optimizing a function of 2 variables.

Let $f(x,y)$ be defined and has continuous first and second partials on a domain $D$. Also, let $$A = \frac{\partial^2 f}{\partial x^2} \\ B = \frac{\partial^2{f}}{\partial x \partial y} \\ C = ...
0
votes
4answers
24 views

Help with Lagrangian Constrained Optimisation

Question: Maximise f (x, y) = x2y, where (x, y) ∈ R2 given the constraint that all (x, y) are points on a circle with radius √3 around origin (0, 0). Solution: f (±√2, 1) = 2 is the maximal value ...
1
vote
2answers
30 views

L1 regularized SVM in Matlab

Minimizing the following SVM formulation \begin{align} \arg\min_{\mathbf{w}}\frac{1}{2}\|\mathbf{w}\|^2_2 \\ \text{subject to } \quad y_i(\mathbf{w}\cdot\mathbf{x_i}) \ge 1 \end{align} can be done ...
0
votes
2answers
1k views

How to plot a phase portrait for this system of differential equations?

I beg your help.. I'd like the phase portrait for this system. I don't know how to use Mathematica/Matlab ... :( If anyone can make this portrait and post a print screen here, I would thank you ...
0
votes
1answer
30 views

Reference for gradient descent with unit norm constraint

I faced a non-convex optimization problem with unit norm constraint. I can solve the problem using the gradient descent method and the projection of the gradient onto the tangent plane as in @joriki ...
1
vote
0answers
23 views

Unconstrained optimization problem (lasso with modification)

I am looking to solve the following unconstrained optimization problem: $$\arg \min_U \|b-A(UY^*)\|_F^2+\lambda\|U\|_1$$ where $\|.\|_F$ is frobenius norm. I know that the solution without the ...
0
votes
1answer
19 views

Optimum set partitioning with constraint

Be $A \subset D \wedge m \in D \wedge \forall x \in A:x < m$, with $D$ finite and included in the positive integers, I need to partition $A$ into $B_n$, while minimizing $n$, so that ...
1
vote
2answers
29 views

multi-objective optimization

I am currently encounterring a optimization problem. The goal is optimize an objective function A and B at the same time. But the problem is that optmizing A will almost always tradoff with B, such ...
0
votes
2answers
262 views

Optimal Basic Feasible Solutions

In linear programming, is it true that you can only have at most 2 optimal basic feasible solutions? If so, why?
0
votes
1answer
16 views

Max/Min Notation Question

In a paper I'm currently reading it gives alpha to be the following value. $\alpha = \max_t \min_{t_j \in T_N} ||t-t_j||_2$ I am wondering what exactly this means? I have the following code: ...
-1
votes
1answer
28 views

determine the maximum cross‐sectional area.

The client wants to maximise the volume of a materials store to be constructed next to a 3  metre high stone wall (shown as OA in the cross section in the diagram). The roof (AB) and  front (BC) are ...
0
votes
1answer
29 views

Minimal volume of a tetrahedral

I'm unsure how to solve the following problem: Let $\textbf{p}=(a,b,c) \in \mathbb{R}^{3}$ with $a,b,c > 0$. For $\alpha , \beta > 0$ the equation $$\alpha (x-a)+ \beta(y-b) + (z-c) =0$$ ...
0
votes
1answer
23 views

Above what order of magnitude a pure cutting-plane algorithm must be forgotten in favour of branch-and-cut?

Crawling the web on the subject of the cutting-plane algorithm, I have seen everywhere that a pure cutting-plane method cannot be used for numerical instability reasons after some iterations. But do ...