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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solution of a system of nonlinear equations

Do you know any method to solve the following system of nonlinear equations ? $\begin{equation} 141,3829=A+\frac{B}{323}+5,78C+F323^{E}\\ 69,07645=A+\frac{B}{333}+5,81C+F333^{E}\\ ...
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16 views

Smallest grammar problem on a single character.

Let the alphabet be $\Sigma = \{a\}$. Say $s = a^6 = aaa aaa$. If the repeated variable $A = aa$ appears $k$ times in the expanded starting rule of a smallest grammar $G_s$ for $s$. Then that ...
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Optimization for nonlinear function with linear constraints

I am trying to come up with nice solution for a specific RCPSP. Context There is a small factory which can produce P kinds of products on N machines each machine is universal but can only create ...
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1answer
26 views

Distributed Convex Optimization Algorithm

Consider the convex optimization problem $$ \min_{x_1, \cdots, x_N, y} \sum_{i=1}^{N} f_i(x_i,y) $$ $$ \text{subject to: } x_i \in X_i \ \ \forall i, \ \ y \in Y, \ \ y = \sum_{i=1}^{N} x_i $$ ...
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Optimization with probabilities

How does one take care of constraints when setting up an expected returns maximization model of the following kind: $y=f(x-r)$ if $z > a$, $z=g(x-s)$ if $w>b$. I want to max $E=prob\times y + ...
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2answers
30 views

How to find local maximum of the function $f(x) = x^3-9x^2+24x+4$?

Give the value of x where the function $f(x) = x^3-9x^2+24x+4$ has a local maximum. a) -4 b) 4 c) 2 d) 3 e) -2 I graphed it and I'm not sure how to find the local max
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1answer
22 views

Reentrant constraints in active set algorithm?

Problem definition Supposing you're trying to solve a quadratic program: $$ \min_x f(x) = \frac{1}{2}x^T Q x + c^T x \\ \mbox{s.t} \, \; A x \ge b$$ Where Q is square ($n$x$n$), positive semi ...
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35 views

Linear programming - Textbook recommendations

Next term, I will attend a course on linear programming. Due to the assignments, we will have to write many thorough proofs. I anticipate that we will be supposed to cope with in-depth background ...
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17 views

How to attack solving for similarity transformed quantities

I'm interested in solving equations of the form: $$ R\mathbf{x}R^{T}=\alpha\mathbf{x}+k $$ where $R$ is a orthonormal matrix (rotation), $\alpha$ is a scalar multiplier (non-zero), ...
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interior point methods on linear programming with matlab [on hold]

I use matlab program language during which I write the block matrix : lhs = [zeros(5,5),A.',eye(5);... A,zeros(3,3),zeros(5,3);... diag(z),zeros(5,5),diag(x)], such that A,z,x are ...
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Quadratic programming with linear equality constraints in Matlab [on hold]

I have to identify an ARX under some linear constraints, this means that I have a quadratic programming with linear equality constraints problem. One way is to use the following equations in the ...
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interior point in linear programming with matlab [on hold]

I use matlab program language during which I write the block matrix : lhs = [zeros(5,5),A.',eye(5);... A,zeros(3,3),zeros(5,3);... diag(z),zeros(5,5),diag(x)], such that A,z,x are ...
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0answers
19 views

Non-convex constraint made cost

Consider the non-convex optimization problem $$ \min_{x \in X} \ f(x) \quad \text{s.t.:} \ \ g(x) \leq 0, \ h(x) = 0 $$ where $X \subset \mathbb{R}^{2n}$ is compact and convex, $f$ and $g$ are ...
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27 views

Teacher/student exam assignment matching problem - equivalent problem?

I have a sort of matching problem. I am wondering if you know if this problem reduces to a familiar one. It arises from my friend's job, and something we were wondering about this morning on the ...
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Which matrix norm gives the minimal variation of eigenvalues?

This is a follow-up of this question. The original question is intentionally as general as possible, because I was interested in the most general possible answer. I am now trying to understand its ...
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39 views

Smallest value taken by a quadratic polynomial in two variables.

Let $p$ be a degree $2$ polynomial with integer coefficients, say $$p(x,y) = Ax^2 + By^2 + Cxy + Dx + Ey + F.$$ I would like to find an algorithm which solves the following: Problem 1: Given ...
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Equivalent optimization problems?

I am wondering if the set of optimizers of the problem $$ \min_{x \in X} \ f(x) \quad \text{subject to: } g(x) \leq 0, \ h(x) = 1 $$ is the same of the one of $$ \min_{x \in X} \ f(x) + h(x) \quad ...
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23 views

Multivariable Gradient Descent

If I have a function $S: \mathbb{R}^2 \to \mathbb{R}$ that describes energy falloff in space. I have a source $S$ positioned at $(S_x, S_y)$ and the intensity at any given point in space (x, y) is ...
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33 views

Min and max of a product.

Let $x_i\in X_i\subset \mathbb{R}$ so that each $X_i$ is a compact set with no isolated points for all $1\leq i\leq n$. Let: $a_i\in X_i$ so that $|x_i|\leq|a_i|$ for all $x_i\in X_i$. $b_i = ...
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Solving bi-linear programming in MATLAB [on hold]

Can anyone suggest any solver in MATLAB to solve a bilinear programming problem? Or any tutorial for the same?
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1answer
28 views

Optimization with a constraint given by a differential equation

I have the following differential equation $$\ddot\theta(t) = -k\sin{\omega t}\sin{\theta(t)} \quad \text{where} \quad \theta(0)=\theta_0, \dot\theta(0)=v_0$$ where $\omega$ is a known constant and ...
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1answer
16 views

Least surface of volume with constraints

We know that in 2D/3D the shape with the least surface of a certain volume is a circle/sphere (e.g. soap bubbles). Now Imagine we have a flat surface (tabletop) that can be used as part of the surface ...
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45 views

Finding Critical Points - Two points on a parabola st joining line is minimised

I have that two points A,B lie on a parabola y = x^2 such that the line segment between them is always perpendicular to the tangent line at A's position. A sits at (a,a^2). Firstly, I found the slope ...
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Comparing two circularly shifted matrices

I am looking for a way to compare two matrices A and B where B is the result of circularly shifting rows of A i.e. A = [1 2 3;4 5 6], B = [4 5 6;1 2 3] Is there an operator or metric that would ...
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3answers
60 views

Interesting, unusual max/min problems?

So I've got to that stage of my elementary mathematics subject for engineers when we talk about differentiation and solution of max/min problems. And I'd like to entertain and engage the students ...
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When is $\min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))$?

When is $$ \min_{x\in X,y\in Y}f(x,y)=\min_{x\in X}(\min_{y\in Y}f(x,y))? $$
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2answers
38 views

Augmented Lagrangian

Consider the following equality constraint minimization problem: minimize $\text{ }f(x)$ subject to $Ax=b$ Its Lagrangian is then: $L(x,y) = f(x) + y^T(Ax-b)$ We can use then gradient ascent to ...
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k- maximally link disjoint paths and equations

This problem is NP-complete and also discussed to some extent in Graph problems which are NP-Complete on directed graphs but polynomial on undirected graphs from the level of my reading from various ...
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1answer
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Matlab: need help with optimization

I am trying to minimize the objective function over [x(1),x(2)]: exp(x(1))*(4*x(1)^2+2*x(2)^2+4*x(1)*x(2)+2*x(2)+1)+b subject to constraint ...
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Finding the widest angle to shoot a soccer ball from the sideline using optimization!! [duplicate]

I'm trying to do an independent project for my Math class, but I was stuck and couldn't figure out how to use optimization to find position along the sideline that gives the widest angle to shoot. ...
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Concave Quadratic Program

Let $X \subset \mathbb{R}^n$ be compact and convex. Consider $$ x^* := \arg\min_{x \in X} x^\top Q x + c^\top x $$ where $Q \prec 0$. I am wondering if there are cases where $x^*$ can be written as ...
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Maximum area of a rectangular field that can be fenced and divided in half by a fence

A rectangular pasture is to be fenced then divided in half by a fence parallel to 2 opposite sides. If a total of 6000m of fencing is used, what is the maximum area that can be fenced? I have no idea ...
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3answers
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Optimizing an expression containing sum of square roots of squared terms

For optimization problems involving square root, it is common to optimize the squared expression instead of that containing the square root. What if we have sum of squared expressions ? Consider the ...
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1answer
20 views

Does existence of global minimum imply coercivity?

It is known that a coercive function over a closed, unbounded set has a global minimum. Is the converse true ? The larger context for this question is the following question: Suppose we are given a ...
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Using bordered Hessian matrix to determine non-degeneracy and type of constrained extremum

I have the following problem: $\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}\def\g{g(x_1,x_2,x_3)}\def\l{\lambda}\def\q{\begin{pmatrix}}\def\p{\end{pmatrix}}$ Find the ...
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Local minimum of the function:

Find the local minimum of the function: $$\def\f{f(x_1,x_2,x_3)}\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\n{\nabla}$$ $$\f=\1^2-2\1\2+2\2^2+\3^2 \text{ in } \mathbb{R}^3$$ $\n\f=(2\1-2\2,-2\1+4\2,2\3) ...
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Write down the HJB equation

Suppose that we have to solve the following optimal control problem \begin{align} V(t,x) = \min_{\alpha}\mathbb{E} \left[\int_{0}^{T}L(t,x,\alpha)dt + F(e^{-\beta t}X^{\alpha}_{T})\right] ...
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Hint for KKT Optimization problem

Can anyone help me with the following optimization problem please? I have to find the $\max f(c,y_1^1,\cdots,y_{N-1}^1,\cdots,y_1^M,\cdots,y_{N-1}^M)=c$ subject to the constraints ...
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1answer
20 views

Distribute N items in K sets with minimum overlap

I am working on an optimization problem to distribute N distinct items (each of the items is available in infinite quantity), among K sets. Each set should have T items. (The constraint of T can be ...
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Using EM versus estimating all parameters directly

My question is really a request for a clarification, and pertains to whether or not there are questions for which it is necessary to use EM or if it's more of a convenience. Let's presume we have n ...
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Regression with error coming from rounding

I am looking at the following model: $c$ is a fixed vector in $\mathbb{R}_+^n$ and for any $x \in \mathbb{R}_+^n$ we obtain a value $y =[c^Tx]$, i.e. rounding $c^Tx$ to the nearest integer. I want ...
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Minimum of summed sequence

Define M non-negative sequences, \begin{equation} a_{m,1}\geq a_{m,2}\geq,...,\geq a_{m,K}\quad \text{for}\ m=1,..,M \end{equation} and cyclic shifted versions $a^{\zeta_m}_{m,k}$ with shift value ...
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Maximizing the profit of a monopolist, given the joint cost function and demand functions for two products

A monopolist produces quantities x and y of two goods, X and Y , respectively and the inverse demand functions for these are given by $$p_X =4−2x \ \text{ and } \ p_Y =2−2y$$ where $p_X$ and ...
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Definition of Global Convergence

I am confused by the notion of "global convergence" as used in numerical optimization literature, and did not find an exact definition for that yet. Now I try to double check my understanding here. ...
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Farthest vector direction relative to other vectors

I wished to know the cheapest computational means (be it analytical or numerical) to find the vector from origin (normalised or not; I do not care about its magnitude) given any arbitrary set of ...
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1answer
31 views

minimax vs max-min

I am a beginner in optimization and I have the following questions: First question: is there a difference between the two optimization formulations? OP1: $\max_{x} \min_{k} g_k=f(x)$ and OP2: ...
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$|p- \dfrac xn|>|q- \dfrac xn|$ $\implies$ $p^x(1-p)^{n-x}<q^x(1-q)^{n-x}$?

If $p,q \in (0,1)$ , and $ n \in \mathbb N$ be given and $x$ be given integer between $0$ and $n$ such that $|p- \dfrac xn|>|q- \dfrac xn|$ , then is it true that ...
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Sufficient conditions for the objective function to have gradient pointing towards the origin

Say I have a sufficiently smooth objective function $J(x)$. How can I ensure the below statement is fulfilled -with additional assumptions to $J$ if needed. $\underset{{{| x ...
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portfolio optimisation

I'm currently implementing a CAPM model in Excel based on the following criteria/features: A portfolio of n risky assets when n=6 (in this case) A riskless borrowing rate of 8% and riskless lending ...