text stringlengths 1 1k | source stringlengths 31 152 |
|---|---|
ISBN 0-486-43227-0.
Bazaraa, Mokhtar S. and Shetty, C. M. (1979). Nonlinear programming. Theory and algorithms. John Wiley & Sons. ISBN 0-471-78610-1.
Bonnans, J. Frédéric; Gilbert, J. Charles; Lemaréchal, Claude; Sagastizábal, Claudia A. (2006). Numerical optimization: Theoretical and practical aspects. Universitext ... | https://en.wikipedia.org/wiki/Nonlinear_programming |
ematical Programming Glossary | https://en.wikipedia.org/wiki/Nonlinear_programming |
Minification (also minimisation or minimization) is the process of removing all unnecessary characters from the source code of interpreted programming languages or markup languages without changing its functionality. These unnecessary characters usually include whitespace characters, new line characters, comments, and ... | https://en.wikipedia.org/wiki/Minification_(programming) |
ll as with the minified source.
The goals of minification are not the same as the goals of obfuscation; the former is often intended to be reversed using a pretty-printer or unminifier. However, to achieve its goals, minification sometimes uses techniques also used by obfuscation; for example, shortening variable names... | https://en.wikipedia.org/wiki/Minification_(programming) |
llowed by YUI Compressor in 2007. In 2009, Google opened up its Closure toolkit, including Closure Compiler which contained a source mapping feature together with a Firefox extension called Closure Inspector. In 2010, Mihai Bazon introduced UglifyJS, which was superseded by UglifyJS2 in 2012; the rewrite was to allow f... | https://en.wikipedia.org/wiki/Minification_(programming) |
reated by Joseph Schorr as part of the Closure Inspector minification project. Version 2 and 3 of the format reduced the size of the map files considerably.
== Types ==
=== Tools ===
Visual Studio Code comes with minification support for several languages. It can readily browse the Visual Studio Marketplace to down... | https://en.wikipedia.org/wiki/Minification_(programming) |
iles into a single file for client download. JavaScript source maps can make code readable and debuggable even after it has been combined and minified.
== References == | https://en.wikipedia.org/wiki/Minification_(programming) |
Verse is a static typed object-oriented programming language created by Epic Games. It was released alongside UEFN in March 2023 and was authored by a team of well-known programmers led by Simon Peyton Jones, and Epic Games CEO Tim Sweeney. Verse is designed to interact with Fortnite Creative's existing devices syste... | https://en.wikipedia.org/wiki/Verse_(programming_language) |
nto scripts if needed.
=== Classes and objects ===
Verse employs a class-based object-oriented programming model. Developers can define classes to represent various game entities and behaviors.
For example:
hello_world_device := class(creative_device):
OnBegin<override>()<suspends>:void=
Print("Hello, wo... | https://en.wikipedia.org/wiki/Verse_(programming_language) |
ific location.
IsLookingAtLocation(Character:fort_character, LocationToCheck: vector3, Threshold: float):logic
=== Event handling ===
The language supports event-driven programming, where events like EliminatedEvent and PlayerAddedEvent can trigger specified functions. Events can also be subscribed to and listened f... | https://en.wikipedia.org/wiki/Verse_(programming_language) |
data. In lambda calculus, functions are first-class citizens, meaning they can be passed as arguments to other functions, returned as values from other functions, and assigned to variables; while in Verse functions can be passed around and manipulated similarly, showcasing the functional programming paradigm. Examples... | https://en.wikipedia.org/wiki/Verse_(programming_language) |
h are defined declaratively.
== Future plans ==
Epic Games has described Verse as the programming language for the metaverse and has factored mass scalability into its development. Until the proposed convergence with Unreal Engine, it is yet to be seen how Verse will behave interacting with various systems and proces... | https://en.wikipedia.org/wiki/Verse_(programming_language) |
Cargo cult programming is a style of computer programming characterized by the ritual inclusion of code or program structures that serve no real purpose. Cargo cult programming is symptomatic of a programmer not understanding either a bug they were attempting to solve or the apparent solution (compare shotgun debugging... | https://en.wikipedia.org/wiki/Cargo_cult_programming |
mology ==
Cargo cults are millenarian movements that arose in Melanesia under colonial rule, and inspired Richard Feynman to coin the expression cargo cult science. In Feynman's description, after the end of the Second World War practitioners believed that air delivery of cargo would resume if they carried out the pro... | https://en.wikipedia.org/wiki/Cargo_cult_programming |
lated term to cargo cult programming in software engineering is cargo cult software engineering, coined by Steve McConnell.: 23-26
McConnell describes software development organizations that attempt to emulate more successful development houses, either by slavishly following a software development process without unde... | https://en.wikipedia.org/wiki/Cargo_cult_programming |
f successful software development organizations) are in fact engaging in what he calls cargo cult software engineering.: 11–13
== See also ==
Black box
Cargo cult science
Copy-and-paste programming
GitHub Copilot
Magic (programming)
Magical thinking
Vibe coding
== References ==
== Further reading == | https://en.wikipedia.org/wiki/Cargo_cult_programming |
In the field of mathematical optimization, stochastic programming is a framework for modeling optimization problems that involve uncertainty. A stochastic program is an optimization problem in which some or all problem parameters are uncertain, but follow known probability distributions. This framework contrasts with d... | https://en.wikipedia.org/wiki/Stochastic_programming |
teger programming for problems in which some variables must be integers
Chance constrained programming for dealing with constraints that must be satisfied with a given probability
Stochastic dynamic programming
Markov decision process
Benders decomposition
== Two-stage problem definition ==
The basic idea of two-stag... | https://en.wikipedia.org/wiki/Stochastic_programming |
(
x
,
ξ
)
]
}
{\displaystyle \min _{x\in X}\{g(x)=f(x)+E_{\xi }[Q(x,\xi )]\}}
where
Q
(
x
,
ξ
)
{\displaystyle Q(x,\xi )}
is the optimal value of the second-stage prob... | https://en.wikipedia.org/wiki/Stochastic_programming |
min
x
∈
R
n
... | https://en.wikipedia.org/wiki/Stochastic_programming |
subject to
A
x
=
b
x
≥
0
... | https://en.wikipedia.org/wiki/Stochastic_programming |
m
q
(
ξ
)
T
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle {\begin{array}{llr}\min \limits _{y\in \mathbb {R} ^{m}}&q(\xi )^{T}y&\\{\text{subject to}}&T(\xi )x+W(\xi )y=h(\xi )&\\&y\geq 0&\end{array}}}
In such formulation
x
∈
R
... | https://en.wikipedia.org/wiki/Stochastic_programming |
irst stage we have to make a "here-and-now" decision
x
{\displaystyle x}
before the realization of the uncertain data
ξ
{\displaystyle \xi }
, viewed as a random vector, is known. At the second stage, after a realization of
... | https://en.wikipedia.org/wiki/Stochastic_programming |
solution as a recourse action where the term
W
y
{\displaystyle Wy}
compensates for a possible inconsistency of the system
T
x
≤
h
{\displaystyle Tx\leq h}
and
q
... | https://en.wikipedia.org/wiki/Stochastic_programming |
above two-stage problem assumes that the second-stage data
ξ
{\displaystyle \xi }
is modeled as a random vector with a known probability distribution. This would be justified in many situations. For example, the distribution of
ξ
{\disp... | https://en.wikipedia.org/wiki/Stochastic_programming |
e often needs to assume that the random vector
ξ
{\displaystyle \xi }
has a finite number of possible realizations, called scenarios, say
ξ
1
,
…
,
ξ
... | https://en.wikipedia.org/wiki/Stochastic_programming |
k
=
1
K
p
k
Q
(
x
,
ξ
k
)
{\displaystyle E[Q(x,\xi )]=\sum ... | https://en.wikipedia.org/wiki/Stochastic_programming |
Optimizers such as CPLEX, and GLPK can solve large linear/nonlinear problems. The NEOS Server, hosted at the University of Wisconsin, Madison, allows free access to many modern solvers. The structure of a deterministic equivalent is particularly amenable to apply decomposition methods, such as Benders' decomposition or... | https://en.wikipedia.org/wiki/Stochastic_programming |
t
h
{\displaystyle k^{th}}
two-period LP, representing the
k
t
h
{\displaystyle k^{th}}
scenario, may be regarded as having the following fo... | https://en.wikipedia.org/wiki/Stochastic_programming |
k
T
z
k
... | https://en.wikipedia.org/wiki/Stochastic_programming |
+
W
k
z
k
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle {\begin{array}{lccccccc}{\text{Minimize}}&f^{T}x&+&g^{T}y&+&h_{k}^{T}z_{k}&&\\{\text{subject to}}&Tx&+&Uy&&&=&r\\&&&V_{k}y&+&W_{k}z_{k}&=&s_{k}\\&x&,&y&,&z_{k}&\geq &0\end{array}}}
The vectors
x
{\disp... | https://en.wikipedia.org/wiki/Stochastic_programming |
iffer in some respects from scenario to scenario, reflecting uncertainty about the future.
Note that solving the
k
t
h
{\displaystyle k^{th}}
two-period LP is equivalent to assuming the
... | https://en.wikipedia.org/wiki/Stochastic_programming |
equivalent. (Strictly speaking a deterministic equivalent is any mathematical program that can be used to compute the optimal first-stage decision, so these will exist for continuous probability distributions as well, when one can represent the second-stage cost in some closed form.)
For example, to form the determini... | https://en.wikipedia.org/wiki/Stochastic_programming |
f
⊤
x
+
g
⊤
... | https://en.wikipedia.org/wiki/Stochastic_programming |
2
h
2
T
z
2
... | https://en.wikipedia.org/wiki/Stochastic_programming |
subject to
T
x
+
U
y
... | https://en.wikipedia.org/wiki/Stochastic_programming |
z
1
=
s
... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
s
2
... | https://en.wikipedia.org/wiki/Stochastic_programming |
+
W
K
z
K
=
... | https://en.wikipedia.org/wiki/Stochastic_programming |
2
,
…
,
z
... | https://en.wikipedia.org/wiki/Stochastic_programming |
}}}
We have a different vector
z
k
{\displaystyle z_{k}}
of later-period variables for each scenario
k
{\displaystyle k}
. The first-period variables
x... | https://en.wikipedia.org/wiki/Stochastic_programming |
experts' opinions on the future. The number of constructed scenarios should be relatively modest so that the obtained deterministic equivalent can be solved with reasonable computational effort. It is often claimed that a solution that is optimal using only a few scenarios provides more adaptable plans than one that a... | https://en.wikipedia.org/wiki/Stochastic_programming |
ns
d
{\displaystyle d}
independent random components, each of which has three possible realizations (for example, future realizations of each random parameters are classified as low, medium and high), then the total number of scenarios is
K
=
... | https://en.wikipedia.org/wiki/Stochastic_programming |
nte Carlo simulation. Suppose the total number of scenarios is very large or even infinite. Suppose further that we can generate a sample
ξ
1
,
ξ
2
,
…
... | https://en.wikipedia.org/wiki/Stochastic_programming |
q(x)=E[Q(x,\xi )]}
is approximated by the sample average
q
^
N
(
x
)
=
1
... | https://en.wikipedia.org/wiki/Stochastic_programming |
N
(
x
)
=
min
x
... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
1
N
Q
(
x
,
ξ
j
... | https://en.wikipedia.org/wiki/Stochastic_programming |
}_{N}(x)=&\min \limits _{x\in \mathbb {R} ^{n}}&c^{T}x+{\frac {1}{N}}\sum _{j=1}^{N}Q(x,\xi ^{j})&\\&{\text{subject to}}&Ax&=&b\\&&x&\geq &0\end{array}}}
This formulation is known as the Sample Average Approximation method. The SAA problem is a function of the considered sample and in that sense is random. For a gi... | https://en.wikipedia.org/wiki/Stochastic_programming |
…
,
N
{\displaystyle j=1,\dots ,N}
, each taken with the same probability
p
j
=
1
N
{\displaystyle p_{j}={\frac {... | https://en.wikipedia.org/wiki/Stochastic_programming |
R
d
{\displaystyle \Xi \subset \mathbb {R} ^{d}}
, and
Q
:
X
×
Ξ
→
R
{\displaystyle Q:X\times \Xi \rightarrow \mathbb {R} }
. In the fr... | https://en.wikipedia.org/wiki/Stochastic_programming |
Q
(
x
,
ξ
)
{\displaystyle Q(x,\xi )}
is finite almost surely.
Suppose that we have a sample
ξ
1
,
…
,
ξ
N
... | https://en.wikipedia.org/wiki/Stochastic_programming |
arity conditions
1
N
∑
j
=
1
N
Q
(
x
,
ξ
j
... | https://en.wikipedia.org/wiki/Stochastic_programming |
N
(
x
)
]
=
g
(
x
)
{\displaystyle E[{\hat {g}}_{N}(x)]=g(x)}
, i.e.,
g
... | https://en.wikipedia.org/wiki/Stochastic_programming |
X
{\displaystyle X}
of the SAA problem is fixed, i.e., it is independent of the sample. Let
ϑ
∗
{\displaystyle \vartheta ^{*}}
and
S
... | https://en.wikipedia.org/wiki/Stochastic_programming |
e {\hat {S}}_{N}}
be the optimal value and the set of optimal solutions, respectively, of the SAA problem.
Let
g
:
X
→
R
{\displaystyle g:X\rightarrow \mathbb {R} }
and
... | https://en.wikipedia.org/wiki/Stochastic_programming |
}
⊂
X
{\displaystyle \{x_{N}\}\subset X}
converging to
x
¯
{\displaystyle {\overline {x}}}
it follows that
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle X}
and
g
^
N
(
⋅
)
{\displaystyle {\hat {g}}_{N}(\cdot )}
converges to
... | https://en.wikipedia.org/wiki/Stochastic_programming |
g(x)}
with probability 1, as
N
→
∞
{\displaystyle N\rightarrow \infty }
, uniformly on the feasible set
X
{\displaystyle X}
. Then
ϑ
... | https://en.wikipedia.org/wiki/Stochastic_programming |
C\subset \mathbb {R} ^{n}}
such that
the set
S
{\displaystyle S}
of optimal solutions of the true problem is nonempty and is contained in
C
{\displaystyle C}
the function
g
(
x
)
... | https://en.wikipedia.org/wiki/Stochastic_programming |
\displaystyle N\rightarrow \infty }
, uniformly in
x
∈
C
{\displaystyle x\in C}
for
N
{\displaystyle N}
large enough the set
S
... | https://en.wikipedia.org/wiki/Stochastic_programming |
→
ϑ
∗
{\displaystyle {\hat {\vartheta }}_{N}\rightarrow \vartheta ^{*}}
and
D
(
S
∗
,
... | https://en.wikipedia.org/wiki/Stochastic_programming |
ystyle A}
from set
B
{\displaystyle B}
, defined as
In some situations the feasible set
X
{\displaystyle X}
of the SAA problem is estimated, then the corresponding SAA problem takes the form
where
... | https://en.wikipedia.org/wiki/Stochastic_programming |
et \mathbb {R} ^{n}}
such that
the set
S
{\displaystyle S}
of optimal solutions of the true problem is nonempty and is contained in
C
{\displaystyle C}
the function
g
(
x
)
... | https://en.wikipedia.org/wiki/Stochastic_programming |
ystyle N\rightarrow \infty }
, uniformly in
x
∈
C
{\displaystyle x\in C}
for
N
{\displaystyle N}
large enough the set
S
... | https://en.wikipedia.org/wiki/Stochastic_programming |
x_{N}\in X_{N}}
and
x
N
{\displaystyle x_{N}}
converges with probability 1 to a point
x
{\displaystyle x}
, then
x
∈
X
... | https://en.wikipedia.org/wiki/Stochastic_programming |
ϑ
^
N
→
ϑ
∗
{\displaystyle {\hat {\vartheta }}_{N}\rightarrow \varthe... | https://en.wikipedia.org/wiki/Stochastic_programming |
ξ
1
,
…
,
ξ
N
{\displaystyle \xi ^{1},\dots ,\xi ^{N}}
is i.i.d. and fix a point
x
∈
X
... | https://en.wikipedia.org/wiki/Stochastic_programming |
x
)
{\displaystyle {\frac {1}{N}}\sigma ^{2}(x)}
, where
σ
2
(
x
)
:=
V
a
r
[
Q
(
x
,
ξ
)
]
... | https://en.wikipedia.org/wiki/Stochastic_programming |
riance
σ
2
(
x
)
{\displaystyle \sigma ^{2}(x)}
, written as
N
(
0
,
σ
... | https://en.wikipedia.org/wiki/Stochastic_programming |
^
N
(
x
)
{\displaystyle {\hat {g}}_{N}(x)}
has approximately normal distribution with mean
g
(
x
)
{\displaystyle g(x)... | https://en.wikipedia.org/wiki/Stochastic_programming |
/
2
:=
Φ
−
1
(
1
−
α
/
2
)
{\displaystyle z_{\alpha /2}:=\Phi ^{-1}(1-\alpha... | https://en.wikipedia.org/wiki/Stochastic_programming |
)
{\displaystyle O({\sqrt {N}})}
.
== Applications and examples ==
=== Biological applications ===
Stochastic dynamic programming is frequently used to model animal behaviour in such fields as behavioural ecology. Empirical tests of models of optimal foraging, life-history transitions su... | https://en.wikipedia.org/wiki/Stochastic_programming |
n example from finance of multi-stage stochastic programming.
Suppose that at time
t
=
0
{\displaystyle t=0}
we have initial capital
W
0
{\displaystyle W_{0}}
to ... | https://en.wikipedia.org/wiki/Stochastic_programming |
n
{\displaystyle n}
assets. Let
x
0
=
(
x
10
,
…
,
x
n... | https://en.wikipedia.org/wiki/Stochastic_programming |
0
{\displaystyle \sum _{i=1}^{n}x_{i0}=W_{0}}
should hold.
Consider the total returns
ξ
t
=
(
ξ
1
t
... | https://en.wikipedia.org/wiki/Stochastic_programming |
At time period
t
=
1
{\displaystyle t=1}
, we can rebalance the portfolio by specifying the amounts
x
1
=
(
x
11
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle \xi _{1}}
, i.e.,
x
1
=
x
1
(
ξ
1
)
{\d... | https://en.wikipedia.org/wiki/Stochastic_programming |
t
(
ξ
[
t
]
)
{\displaystyle x_{t}=x_{t}(\xi _{[t]})}
of the available information given by
ξ
[
... | https://en.wikipedia.org/wiki/Stochastic_programming |
ξ
[
t
]
)
{\displaystyle x_{t}=x_{t}(\xi _{[t]})}
,
t
=
0
,
…
,
T
−
1
{\displaystyle t=0,\dots ,T-1}
, with
... | https://en.wikipedia.org/wiki/Stochastic_programming |
i
=
1
,
…
,
n
{\displaystyle i=1,\dots ,n}
,
t
=
0
,
…
,
T
−
1
{\displaystyle t=0,\dots ,T-1}
, and the balance of wealth constraints,... | https://en.wikipedia.org/wiki/Stochastic_programming |
ystyle t=1,\dots ,T}
the wealth
W
t
{\displaystyle W_{t}}
is given by
W
t
=
∑
i
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle t}
.
Suppose the objective is to maximize the expected utility of this wealth at the last period, that is, to consider the problem
max
E
[
U
(
W
T
)
]
... | https://en.wikipedia.org/wiki/Stochastic_programming |
,
ξ
T
{\displaystyle \xi _{1},\dots ,\xi _{T}}
. This can be done in various ways. For example, one can construct a particular scenario tree defining time evolution of the process. If at every stage the random return of each asset ... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
(
ξ
1
,
…
,
ξ
T
−
1
)
{\displaystyle \xi _{[T-1]}=(\xi _{1},\dots ,\xi _{T-1})}
of the random... | https://en.wikipedia.org/wiki/Stochastic_programming |
E
[
U
(
W
T
)
|
ξ
... | https://en.wikipedia.org/wiki/Stochastic_programming |
n
ξ
i
T
x
i
... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
W
T
−
1
x
... | https://en.wikipedia.org/wiki/Stochastic_programming |
)
|
ξ
[
T
−
1
]
]
{\displaystyle E[U(W_{T})|\xi _{[T-1]}]}
denotes the conditional expectation of
U
(
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle \xi _{[T-1]}}
and is denoted
Q
T
−
1
(
W
T
−
1
,
... | https://en.wikipedia.org/wiki/Stochastic_programming |
E
[
Q
t
+
1
(
... | https://en.wikipedia.org/wiki/Stochastic_programming |
subject to
W
t
+
1
... | https://en.wikipedia.org/wiki/Stochastic_programming |
∑
i
=
1
n
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle {\begin{array}{lrclr}\max \limits _{x_{t}}&E[Q_{t+1}(W_{t+1},\xi _{[t+1]})|\xi _{[t]}]&\\{\text{subject to}}&W_{t+1}&=&\sum _{i=1}^{n}\xi _{i,t+1}x_{i,t}\\&\sum _{i=1}^{n}x_{i,t}&=&W_{t}\\&x_{t}&\geq &0\end{array}}}
whose optimal value is denoted by
... | https://en.wikipedia.org/wiki/Stochastic_programming |
0
E
[
Q
1
(
... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
∑
i
=
1
n
ξ
... | https://en.wikipedia.org/wiki/Stochastic_programming |
=
W
0
... | https://en.wikipedia.org/wiki/Stochastic_programming |
{\displaystyle \xi _{t}}
, it may be hard to solve these dynamic programming equations. The situation simplifies dramatically if the process
ξ
t
{\displaystyle \xi _{t}}
is stagewise independent, i.e.,... | https://en.wikipedia.org/wiki/Stochastic_programming |
ecome unconditional expectations, and the function
Q
t
(
W
t
)
{\displaystyle Q_{t}(W_{t})}
,
t
=
1
... | https://en.wikipedia.org/wiki/Stochastic_programming |
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