The formulation
min
x
,
y
f
(
x
)
+
g
(
y
)
,
subject
to
A
x
+
B
y
=
b
,
where
f
and
g
are extended-value convex functions, arises in many application areas such as signal processing, imaging and ...image processing, statistics, and machine learning either naturally or after variable splitting. In many common problems, one of the two objective functions is strictly convex and has Lipschitz continuous gradient. On this kind of problem, a very effective approach is the alternating direction method of multipliers (ADM or ADMM), which solves a sequence of
f
/
g
-decoupled subproblems. However, its effectiveness has not been matched by a provably fast rate of convergence; only sublinear rates such as
O
(1 /
k
) and
O
(
1
/
k
2
)
were recently established in the literature, though the
O
(1 /
k
) rates do not require strong convexity. This paper shows that global linear convergence can be guaranteed under the assumptions of strong convexity and Lipschitz gradient on one of the two functions, along with certain rank assumptions on
A
and
B
. The result applies to various generalizations of ADM that allow the subproblems to be solved faster and less exactly in certain manners. The derived rate of convergence also provides some theoretical guidance for optimizing the ADM parameters. In addition, this paper makes meaningful extensions to the existing global convergence theory of ADM generalizations.
In this paper, by applying the decision theorem of the Schur-power convex function, the Schur-power convexity of a class of complete symmetric functions are studied. As applications, some new ...inequalities are established.
CVXPY is a domain-specific language for convex optimization embedded in Python. It allows the user to express convex optimization problems in a natural syntax that follows the math, rather than in ...the restrictive standard form required by solvers. CVXPY makes it easy to combine convex optimization with high-level features of Python such as parallelism and object-oriented design. CVXPY is available at http://www.cvxpy.org/ under the GPL license, along with documentation and examples.
•A novel convexity-oriented time-dependent reliability-based topology optimization (CTRBTO) scheme is proposed.•The full-dimensional convex set collocation approach is presented to reveal the convex ...process of displacement responses.•The convex time-dependent reliability (CTR) index is defined and its design sensitivity about design variables is deduced.
In this work, a novel convexity-oriented time-dependent reliability-based topology optimization (CTRBTO) framework is investigated with overall consideration of universal uncertainties and time-varying natures in configuration design. For uncertain factors, the initial static ones are quantified by the convex set model and nodal dynamic responses are then expressed by the convex process model, where both the boundary rules and time-dependency properties are revealed by the full-dimensional convex-set collocation theorem. Unlike the original deterministic constraints in topology optimization schemes, a new convex time-dependent reliability (CTR) index is defined to give a reasonable failure judgment of local dynamic stiffness and impel the overall CTRBTO strategy. In addition, the gradient-based iterative algorithm is utilized to guarantee the computational robustness and the CTR-driven design sensitivities are explicitly analyzed by the Lagrange multiplier method. Several numerical examples are used to illustrate the effectiveness of the proposed method, and numerical results reflect the significance of this study to a certain extent.
We investigate Sobolev spaces W1,Φ associated to Musielak-Orlicz spaces LΦ. We first present conditions for the boundedness of the Voltera operator in LΦ. Employing this, we provide necessary and ...sufficient conditions for W1,Φ to contain isomorphic subspaces to ℓ∞ or ℓ1. Further we give necessary and sufficient conditions in terms of the function Φ or its complementary function Φ⁎ for reflexivity, uniform convexity, B-convexity and superreflexivity of W1,Φ. As corollaries we obtain the corresponding results for Orlicz-Sobolev spaces W1,φ where φ is an Orlicz function, the variable exponent Sobolev spaces W1,p(⋅) and the Sobolev spaces associated to double phase functionals.
In the paper Jónás et al. (2022) 1 we asserted a sufficient condition for the concavity (convexity) of the tau function, which is a generator function-based parametric mapping. In our original ...definition, the generators of the tau function are all the additive generators of the strict triangular norms and strict triangular conorms. We realized that with the original definition, our assertion regarding the concavity (convexity) of the tau function is not valid for all its generators. In this erratum, we present a corrected definition for the tau function such that with this new definition the original assertion regarding the concavity (convexity) of the tau function is valid. With the new definition for the tau function, on the one hand, we narrow the class of its generators, but on the other hand, this class still remains quite versatile and all the other results of the referenced paper remain valid.
Convexity preserving condition of subdivision curve is the focus of computer aided geometric design. In this note, we point out that the four point interpolation subdivision curve is everywhere ...non-convex or non-concave unless it contains a polynomial curve of degree three or less.
Convexity preserving condition of subdivision curve is the focus of computer aided geometric design and has been discussed extensively. In this note, we point out that the well-known four point interpolation subdivision curve is everywhere non-convex or non-concave unless it contains a cubic or less polynomial curve.
Physics-constrained data-driven computing is an emerging hybrid approach that integrates universal physical laws with data-driven models of experimental data for scientific computing. A new ...data-driven simulation approach coupled with a locally convex reconstruction, termed the local convexity data-driven (LCDD) computing, is proposed to enhance accuracy and robustness against noise and outliers in data sets in the data-driven computing. In this approach, for a given state obtained by the physical simulation, the corresponding optimum experimental solution is sought by projecting the state onto the associated local convex manifold reconstructed based on the nearest experimental data. This learning process of local data structure is less sensitive to noisy data and consequently yields better accuracy. A penalty relaxation is also introduced to recast the local learning solver in the context of non-negative least squares that can be solved effectively. The reproducing kernel approximation with stabilized nodal integration is employed for the solution of the physical manifold to allow reduced stress–strain data at the discrete points for enhanced effectiveness in the LCDD learning solver. Due to the inherent manifold learning properties, LCDD performs well for high-dimensional data sets that are relatively sparse in real-world engineering applications. Numerical tests demonstrated that LCDD enhances nearly one order of accuracy compared to the standard distance-minimization data-driven scheme when dealing with noisy database, and a linear exactness is achieved when local stress–strain relation is linear.
•A new physics-based data-driven approach is proposed for elastic solids.•Locally convex material manifolds are constructed to deal with noisy databases.•Accuracy and smoothness of physical states are enhanced by using meshfree methods.•Manifold learning achieves the locally linear exactness with dimension reduction.•Improved convergence, accuracy and robustness against noisy data are demonstrated.
Iterative approaches have been established to be fundamental for the creation of fractals. This paper introduces an approach to visualize Julia and Mandelbrot sets for a complex function of the form ...Q(z)=zp+logct for all z∈ℂ, where p∈N∖{1},t∈1,∞),c∈ℂ∖{0}, using a four-step iteration scheme extended with s-convexity. The study introduces an escape criteria for generating Julia and Mandelbrot sets using a four-step iterative method. It investigates how changes in the iteration parameters influence the shape and color of the resulting Julia and Mandelbrot sets. This approach can generate a wide range of captivating fractals and analyze them through numerical experiments.
A set is detour monophonic convexif The detour monophonic convexity number is denoted by is the cardinality of a maximum proper detour monophonic convex subset of Some general properties satisfied by ...this concept are studied. The detour monophonic convexity number of certain classes of graphs are determined. It is shown that for every pair of integers and with there exists a connected graph such that and , where is the monophonic convexity number of G
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