Mathematical Programming Computation
SCIE-ISI SCOPUS (2009-2023)
1867-2949
1867-2957
Đức
Cơ quản chủ quản: Springer Verlag , Springer Heidelberg
Lĩnh vực:
SoftwareTheoretical Computer Science
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A quasi-Newton algorithm for nonconvex, nonsmooth optimization with global convergence guarantees
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On fast trust region methods for quadratic models with linear constraintsAbstract
Quadratic models
$$Q_k ( \underline{x}), \underline{x}\in \mathcal{R}^n$$
Q
k
(
x
̲
)
,
x
̲
∈
R
n
, of the objective function
$$F ( \underline{x}), \underline{x}\in \mathcal{R}^n$$
F
(
x
̲
)
,
x
̲
∈
R
n
, are used by many successful iterative algorithms for minimization, where k is the iteration number. Given the vector of variables
$$\underline{x}_k \in \mathcal{R}^n$$
x
̲
k
∈
R
n
, a new vector
$$\underline{x}_{k+1}$$
x
̲
k
+
1
may be calculated that satisfies
$$Q_k ( \underline{x}_{k+1} ) < Q_k ( \underline{x}_k )$$
Q
k
(
x
̲
k
+
1
)
<
Q
k
(
x
̲
k
)
, in the hope that it provides the reduction
$$F ( \underline{x}_{k+1} ) < F ( \underline{x}_k )$$
F
(
x
̲
k
+
1
)
<
F
(
x
̲
k
)
. Trust region methods include a bound of the form
$$\Vert \underline{x}_{k+1} - \underline{x}_k \Vert \le \Delta _k$$
‖
x
̲
k
+
1
-
x
̲
k
‖
≤
Δ
k
. Also we allow general linear constraints on the variables that have to hold at
$$\underline{x}_k$$
x
̲
k
and at
$$\underline{x}_{k+1}$$
x
̲
k
+
1
. We consider the construction of
$$\underline{x}_{k+1}$$
x
̲
k
+
1
, using only of magnitude
$$n^2$$
n
2
operations on a typical iteration when n is large. The linear constraints are treated by active sets, which may be updated during an iteration, and which decrease the number of degrees of freedom in the variables temporarily, by restricting
$$\underline{x}$$
x
̲
to an affine subset of
$$\mathcal{R}^n$$
R
n
. Conjugate gradient and Krylov subspace methods are addressed for adjusting the reduced variables, but the resultant steps are expressed in terms of the original variables. Termination conditions are given that are intended to combine suitable reductions in
$$Q_k ( \cdot )$$
hiện toàn bộ
Tập 7 Số 3 - Trang 237-267 - 2015