Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [optimization]

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.

In mathematics, computer science, economics, or management science, mathematical optimization (alternatively, optimization or mathematical programming) is the selection of a best element (with regard to some criteria) from some set of available alternatives.

An optimization problem can be represented in the following way: given a function $f:A\to\mathbb{R}$ from some set $A$ to the real numbers, we want to find an element $x_0\in A$ such that $f(x_0)\le f(x)$ for all $x \in A$ ("minimization") or such that $f(x_0)\ge f(x)$ for all $x \in A$ ("maximization").

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An algorithm for arranging students around a classroom

I am trying to design an algorithm to optimally arrange students around a classroom in defined seats. For each possible pair of students, I have calculated a compatibility score. Desks are not necessarily arranged uniformly, so the geometric…
Rob
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Do constraints make an optimization problem harder or easier?

Assuming that constraints are satisfiable. In general, does adding constraints make an optimization problem harder since the optimal solution does not necessarily satisfy all constraints? Or does it make it easier by reducing the search space for a…
Rufus
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Tricky algebra for minimization

Find the local minimum for $f(x, y) = 2x^4 + y^2 - 4xy + 5y,\:x,y \in \mathbb{R}$ find the local minimum. Okay this seems easy enough, the necessary condition dictates that candidates are of the form $\nabla f(x_0,y_0)=(0,0)$ but $\nabla…
bobdylan
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Is $y=x+\frac{1}{x}$ a hyperbola?

How do we confirm or disprove that? And is there a name for this kind of function? $$f(x)=c(x-a)+\frac{d}{x-a}+b$$ If we restrict that $x-a>0$ and $c,d>0$, an observation is that the minimum $2\sqrt{cd}+b$ is reached when…
SomeShyGuy
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Solving rank-deficient, ill-conditioned linear system

I'm trying to solve a rank-deficient, ill-conditioned linear system. To say, I have a linear system $Ax = b$, where $A$ is an extremely retangular matrix, with size of $49680\times1604$. Rank of the system matrix is $1389$, and the condition number…
HwanHui Jo
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stability of solutions of constrained quadratic optimization problem

I have the following parametric constrained quadratic optimization problem: $\min_{x\in\mathbb{R}^n} \dfrac{1}{2}x^TK(\alpha)x+c(\alpha)^Tx$ subject to $Ax\preceq b$, $Cx=d$ where $\alpha\in[M_1,M_2]$ is the parameter with $0
Ergodic
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Exponential fit - LS

I would like to fit a function of type $y(t) = v_0 (1 - e^{-a \, t})$, for unknown $v_0$ and $a$. I have tried already to obtain the linear form to apply least squares, but without success. Could you enlight me in some way to obtain it without have…
Bruno Lobo
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BFGS in comparison with Gaussian newton

I'm reading about quasi-newton method, and I get the key idea is to find an approximated Hessian Matrix. The most popular one is for sure L-BFGS. But I've got see Gaussian newton method simply estimate the Hessian with $\nabla J \cdot \nabla J^T$.…
Coconut
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Matrix minimization Optimization

When I study the output-based optimal control problem, I meet such a optimization problem as $\min\limits_{K\in \mathbb{R}^{s\times m}} x^TC^T K^T RKCx+x^TBKCx+x^TC^TK^TB^Tx$ where $x\in \mathbb{R}^n$ can be any vector and $R\in \mathbb{R}^{s\times…
peng
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Minimize cost function with constraint

I have an optimization question that I'm not sure if I interpreted correctly. A hospital wants to determine a drug amount in a patient's urine using an instrument/method. The maximum total inaccuracy must be less than 0.25 μg for at least 95% of the…
Nwalk Maple
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maximizing income and quadratic function

The manager of a $1000$ seat concert hall knows from experience that all seats will be occupied if the price of the ticket is $50$ dollars. A market survey indicates that $10$ additional seats will remain empty for each $5$ dollar increase of the…
Keith
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Shopping portfolio question

I have been given a problem phrased in the following terms: John's mother asked him to go to the department store to buy Christmas decorations. Assume that each decoration item has price $p_i$ and utility $u_i$. John has a total budget of $B$ and…
user202531
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Minimizing a function that contains a $\min$

How do I minimize the following objective function? $$f(\theta) = \frac{1}{2}(\theta - z)^2 + \min_{\gamma \geq 0} \left\{\gamma |\theta| + \frac{a}{2}(\gamma - \lambda)^2 \right\}$$ where $z \in \mathbb{R}$, $\lambda > 0$, $a > 1$. I would share my…
user3294195
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Finding optimal variable values where variables are multiplied in pairs and summed

Suppose that I have several $x_i$ variables ($0 \le i \lt N$). And I have equations of the form: $$w_0x_{n_0}x_{n_1} + w_1x_{n_2}x_{n_3} + w_2x_{n_4}x_{n_5} + w_3x_{n_6}x_{n_7} = K_0$$ $$w_4x_{n_8}x_{n_9} + w_5x_{n_{10}}x_{n_{11}} +…
geza
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Is this dynamic optimization?

I would like to know what I should know to understand this IMF paper. What kind of optimization is used to maximize the utility function on page 9 (number 1) subject to constraints (2) and (3)? The function I must maximize…
Luigi
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