- Daftar algoritme
- Sequential quadratic programming
- Sequential linear-quadratic programming
- Successive linear programming
- Quadratic programming
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- Integer programming
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- Quadratically constrained quadratic program
Pulp Fiction (1994)
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Francis R. Mahony III
The Irishman (2019)
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Francisco Ortiz, Martin Scorsese
M3GAN (2022)
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Guy Campbell
Sequential linear-quadratic programming GudangMovies21 Rebahinxxi LK21
Sequential linear-quadratic programming (SLQP) is an iterative method for nonlinear optimization problems where objective function and constraints are twice continuously differentiable. Similarly to sequential quadratic programming (SQP), SLQP proceeds by solving a sequence of optimization subproblems. The difference between the two approaches is that:
in SQP, each subproblem is a quadratic program, with a quadratic model of the objective subject to a linearization of the constraints
in SLQP, two subproblems are solved at each step: a linear program (LP) used to determine an active set, followed by an equality-constrained quadratic program (EQP) used to compute the total step
This decomposition makes SLQP suitable to large-scale optimization problems, for which efficient LP and EQP solvers are available, these problems being easier to scale than full-fledged quadratic programs.
It may be considered related to, but distinct from, quasi-Newton methods.
Algorithm basics
Consider a nonlinear programming problem of the form:
min
x
f
(
x
)
s.t.
b
(
x
)
≥
0
c
(
x
)
=
0.
{\displaystyle {\begin{array}{rl}\min \limits _{x}&f(x)\\{\mbox{s.t.}}&b(x)\geq 0\\&c(x)=0.\end{array}}}
The Lagrangian for this problem is
L
(
x
,
λ
,
σ
)
=
f
(
x
)
−
λ
T
b
(
x
)
−
σ
T
c
(
x
)
,
{\displaystyle {\mathcal {L}}(x,\lambda ,\sigma )=f(x)-\lambda ^{T}b(x)-\sigma ^{T}c(x),}
where
λ
≥
0
{\displaystyle \lambda \geq 0}
and
σ
{\displaystyle \sigma }
are Lagrange multipliers.
= LP phase
=In the LP phase of SLQP, the following linear program is solved:
min
d
f
(
x
k
)
+
∇
f
(
x
k
)
T
d
s
.
t
.
b
(
x
k
)
+
∇
b
(
x
k
)
T
d
≥
0
c
(
x
k
)
+
∇
c
(
x
k
)
T
d
=
0.
{\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d\\\mathrm {s.t.} &b(x_{k})+\nabla b(x_{k})^{T}d\geq 0\\&c(x_{k})+\nabla c(x_{k})^{T}d=0.\end{array}}}
Let
A
k
{\displaystyle {\cal {A}}_{k}}
denote the active set at the optimum
d
LP
∗
{\displaystyle d_{\text{LP}}^{*}}
of this problem, that is to say, the set of constraints that are equal to zero at
d
LP
∗
{\displaystyle d_{\text{LP}}^{*}}
. Denote by
b
A
k
{\displaystyle b_{{\cal {A}}_{k}}}
and
c
A
k
{\displaystyle c_{{\cal {A}}_{k}}}
the sub-vectors of
b
{\displaystyle b}
and
c
{\displaystyle c}
corresponding to elements of
A
k
{\displaystyle {\cal {A}}_{k}}
.
= EQP phase
=In the EQP phase of SLQP, the search direction
d
k
{\displaystyle d_{k}}
of the step is obtained by solving the following equality-constrained quadratic program:
min
d
f
(
x
k
)
+
∇
f
(
x
k
)
T
d
+
1
2
d
T
∇
x
x
2
L
(
x
k
,
λ
k
,
σ
k
)
d
s
.
t
.
b
A
k
(
x
k
)
+
∇
b
A
k
(
x
k
)
T
d
=
0
c
A
k
(
x
k
)
+
∇
c
A
k
(
x
k
)
T
d
=
0.
{\displaystyle {\begin{array}{rl}\min \limits _{d}&f(x_{k})+\nabla f(x_{k})^{T}d+{\tfrac {1}{2}}d^{T}\nabla _{xx}^{2}{\mathcal {L}}(x_{k},\lambda _{k},\sigma _{k})d\\\mathrm {s.t.} &b_{{\cal {A}}_{k}}(x_{k})+\nabla b_{{\cal {A}}_{k}}(x_{k})^{T}d=0\\&c_{{\cal {A}}_{k}}(x_{k})+\nabla c_{{\cal {A}}_{k}}(x_{k})^{T}d=0.\end{array}}}
Note that the term
f
(
x
k
)
{\displaystyle f(x_{k})}
in the objective functions above may be left out for the minimization problems, since it is constant.
See also
Newton's method
Secant method
Sequential linear programming
Sequential quadratic programming
Notes
References
Jorge Nocedal and Stephen J. Wright (2006). Numerical Optimization. Springer. ISBN 0-387-30303-0.
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