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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Max $Z=4x_1+5x_2$ Using two phase simplex

max $$z=4x_1+5x_2$$ s.t $$3x_1+x_2\leq27$$ $$x_1+x_2=12$$ $$3x_1+2x_2\geq 30$$ $$x_1,x_2\geq 0$$ To start the process we find an initial solution, so we add a slack variables: $$3x_1+x_2+x_3 = 27$$ $$x_1+x_2+x_4=12$$ $$3x_1+2x_2-x_5=…
newhere
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Solving two simple optimization problems simultaneously

I have two optimization problems to solve simultaneously. The problems are $$\max_{x}xF(a-x+y)$$ and $$\max_y yF(b-y+x)$$. Here, $a$ and $b$ are scalars and $F$ is a cdf. Is there any technique or easy way that I can use to solve for a solution…
Andeanlll
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glossary explanation of optimal method. Penalty method & Augmented Lagrangian method

In the optimization,i believe lots of people heard about Penalty method & Augmented Lagrangian method,but i wonder why are the creators use "penalty" and "Augmented" to name these method . Penalty means like punishment,but i can't understand the…
electronic component
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How to check whether f is a convex and find all stationary points

Here is the problem What I attempted was calculating the Hessian and trying to prove that it's positive definite/semi-definite or negative definite/semi-definite. It doesn't seam to work as I am getting that the second order principle minor can be…
Ruben Shahnazaryan
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Optimization: Absolut value function

I want to find the minimum of the following function, using a linear solver in Matlab: f = sum(((P * x - d)+|P * x - d|))*0.5*p) x (dimension [ix1]) is binary, P (dimension [nxi]),d and p are always positive. In other words: if P*x => d f =…
Johnny
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How to solve an optimization problem with mean-deviation problem?

I would like to consider how to solve the following optimization problem: $\max x^{T}\mathbf{a}-b\sqrt{x^{T}\Sigma x}\;s.t.x^{T}\mathbf{1}=1$, provided that $b>0$ and $\Sigma$ is positive definite. I want to solve KKT condition to solve the…
will_cheuk
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How do you formulate a "multi-supplier" problem as a MOP?

How do you formulate a "multi-supplier" problem as a MOP (Multi-Objective Optimization Problem)? E.g. a basic linear program where you have some suppliers for resources and you are the purchaser. The goal would be to decide from which suppliers to…
mavavilj
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Differences between RHC and control horizon in MPC

I have used model predictive control strategy in order to optimize a linear discrete SISo model. For example: $x(t+1)=-2x(t)+4u(t)$ Where I want to regulate my state $x(t)$ at the desired point (a simple regulating problem). In every step, I predict…
Seyyed Hossein Ghenaati
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Existence of minimizer of a strongly convex quadratic function on a closed set?

$$ \min\limits_{x\in \mathcal{X}} \|x\|_2^2 $$ where $\mathcal{X}\subseteq R^N$ is a closed set but not necessary convex and can be unbounded. Then existence of optimal solution to this problem holds or not?
zly
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Sufficiency without second derivative test

Consider a function $\mathcal C^1 \ni f : \mathbb R_+ \to \mathbb R$. The following properties are given: \begin{align} &f(0) = 0,\\ &f(\infty) = -\infty,\\ &f'(\bar x) = 0,\\ &|\{x : f'(x) = 0\}| = 1. \end{align} Claim: There exists a unique global…
clueless
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Multi- Objective optimization: how to prove :An optimal solution of the max-ordering problem is weakly efficient but not necessarily efficient.

For multi-objective optimization problem. A method to find weakly efficient is using max-ordering, which is $min_{x\in X}\ max_{i=1,...n}f_i(x)$. How to prove that:An optimal solution $x^o$ of the max-ordering problem $min_{x\in X}\ …
Alan
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Differences between online optimization and realtime optimization

I want to know what differences between online and real-time optimization. I think in real time system disturbance and uncertainty must be considered even in every moment. Any help would be appreciated.
Seyyed Hossein Ghenaati
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Reducing dimensions by considering ex-ante instead of ex-post symmetry

Let $\mathbf x = (x_1,\ldots,x_n) \in \mathbb R^n$. Consider the following problem: \begin{align} \max_{\mathbf x}f(\mathbf x). \end{align} Presume that $\mathbf{\bar x} = \underbrace{(\bar x,\ldots,\bar x)}_{n \text{ times}}$ is the unique…
clueless
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Difference between optimization and reoptimization.

I want to know the main differences between optimization and reoptimization concept. Then can we say model predictive control (MPC) as a rolling horizon control (RHC) method is a reoptimization strategy because it measures outputs and runs online…
Seyyed Hossein Ghenaati
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Multiobjective with single objectives that use distinct components of input vec? What does it mean for other components in input vec?

Multiobjective with single objectives that use distinct components of input vec? What does it mean for other components in input vec? E.g. $$\min (x_1,x_2)$$ $$x \in \mathbb{R}^2, x_1,x_2\geq0$$ Then if one solves this by one objective at a time,…
mavavilj
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