Partial Smoothness Research Articles

On the geometry and refined rate of primal–dual hybrid gradient for linear programming

AbstractWe study the convergence behaviors of primal–dual hybrid gradient (PDHG) for solving linear programming (LP). PDHG is the base algorithm of a new general-purpose first-order method LP solver, PDLP, which aims to scale up LP by taking advantage of modern computing architectures. Despite its numerical success, the theoretical understanding of PDHG for LP is still very limited; the previous complexity result relies on the global Hoffman constant of the KKT system, which is known to be very loose and uninformative. In this work, we aim to develop a fundamental understanding of the convergence behaviors of PDHG for LP and to develop a refined complexity rate that does not rely on the global Hoffman constant. We show that there are two major stages of PDHG for LP: in Stage I, PDHG identifies active variables and the length of the first stage is driven by a certain quantity which measures how close the non-degeneracy part of the LP instance is to degeneracy; in Stage II, PDHG effectively solves a homogeneous linear inequality system, and the complexity of the second stage is driven by a well-behaved local sharpness constant of the system. This finding is closely related to the concept of partial smoothness in non-smooth optimization, and it is the first complexity result of finite time identification without the non-degeneracy assumption. An interesting implication of our results is that degeneracy itself does not slow down the convergence of PDHG for LP, but near-degeneracy does.

On Partial Smoothness, Activity Identification and Faster Algorithms of $L_{1}$ Over $L_{2}$ Minimization

On Partial Smoothness, Activity Identification and Faster Algorithms of $L_{1}$ Over $L_{2}$ Minimization

Partial Smoothness and Constant Rank

The idea of partial smoothness in optimization blends certain smooth and nonsmooth properties of feasible regions and objective functions. As a consequence, the standard first-order conditions guarantee that diverse iterative algorithms (and post-optimality analyses) identify active structure or constraints. However, by instead focusing directly on the first-order conditions, the formal concept of partial smoothness simplifies dramatically: in basic differential geometric language, it is just a constant-rank condition. In this view, partial smoothness extends to more general mappings, such as saddlepoint operators underlying primal-dual splitting algorithms.

Geometric Piecewise Cubic Bézier Interpolating Polynomial with C2 Continuity

Bézier curve is a parametric polynomial that is applied to produce good piecewise interpolation methods with more advantage over the other piecewise polynomials. It is, therefore, crucial to construct Bézier curves that are smooth and able to increase the accuracy of the solutions. Most of the known strategies for determining internal control points for piecewise Bezier curves achieve only partial smoothness, satisfying the first order of continuity. Some solutions allow you to construct interpolation polynomials with smoothness in width along the approximating curve. However, they are still unable to handle the locations of the inner control points. The partial smoothness and non-controlling locations of inner control points may affect the accuracy of the approximate curve of the dataset. In order to improve the smoothness and accuracy of the previous strategies, а new piecewise cubic Bézier polynomial with second-order of continuity C2 is proposed in this study to estimate missing values. The proposed method employs geometric construction to find the inner control points for each adjacent subinterval of the given dataset. Not only the proposed method preserves stability and smoothness, the error analysis of numerical results also indicates that the resultant interpolating polynomial is more accurate than the ones produced by the existing methods.

Active‐Set Newton Methods and Partial Smoothness

Diverse optimization algorithms correctly identify, in finite time, intrinsic constraints that must be active at optimality. Analogous behavior extends beyond optimization to systems involving partly smooth operators, and in particular to variational inequalities over partly smooth sets. As in classical nonlinear programming, such active‐set structure underlies the design of accelerated local algorithms of Newton type. We formalize this idea in broad generality as a simple linearization scheme for two intersecting manifolds.

Partial Smoothness of the Numerical Radius at Matrices Whose Fields of Values are Disks

Solutions to optimization problems involving the numerical radius often belong to the class of “disk matrices”: those whose field of values is a circular disk in the complex plane centered at zero. We investigate this phenomenon using the variational-analytic idea of partial smoothness. We give conditions under which the set of disk matrices is locally a manifold $\mathcal M$, with respect to which the numerical radius $r$ is partly smooth, implying that $r$ is smooth when restricted to $\mathcal M$ but strictly nonsmooth when restricted to lines transversal to $\mathcal M$. Consequently, minimizers of the numerical radius of a parametrized matrix often lie in $\mathcal M$. Partial smoothness holds, in particular, at $n\times n$ matrices with exactly $n-1$ nonzeros, all on the superdiagonal. On the other hand, in the real 18-dimensional vector space of complex $3\times 3$ matrices, the disk matrices comprise the closure of a semialgebraic manifold $\mathcal L$ with dimension 12, and the numerical radius is partly smooth with respect to $\mathcal L$.

The $\mathcal {U}$-Lagrangian, Fast Track, and Partial Smoothness of a Prox-regular Function

When restricted to a subspace, a nonsmooth function can be differentiable. It is known that for a nonsmooth convex function and a point, the Euclidean space can be decomposed into two subspaces: $\mathcal {U}$, over which a special Lagrangian can be defined and has nice smooth properties and $\mathcal {V}$, the orthogonal complement subspace of $\mathcal {U}$. In this paper we generalize the definition of $\mathcal {VU}$-decomposition and $\mathcal {U}$-Lagrangian to prox-regular functions and show that the closely related notions fast track and partial smoothness are equivalent under some conditions. Some connections with tilt stability are discussed.

On partial smoothness, tilt stability and the $${\mathcal {VU}}$$-decomposition

Under the assumption of prox-regularity and the presence of a tilt stable local minimum we are able to show that a $${\mathcal {VU}}$$ like decomposition gives rise to the existence of a smooth manifold on which the function in question coincides locally with a smooth function.

Regularity of solutions of the model Venttsel’ problem for quasilinear parabolic systems with nonsmooth in time principal matrices

The Venttsel’ problem in the model statement for quasilinear parabolic systems of equations with nondiagonal principal matrices is considered. It is only assumed that the principal matrices and the boundary condition are bounded with respect to the time variable. The partial smoothness of the weak solutions (Holder continuity on a set of full measure up to the surface on which the Venttsel’ condition is defined) is proved. The proof uses the A(t)-caloric approximation method, which was also used in [1] to investigate the regularity of the solution to the corresponding linear problem.

Adaptive regularization-based super resolution reconstruction technique for multi-focus low-resolution images

Super Resolution (SR) is the process of enhancing the visual quality of a sequence of observed low-resolution images by constructing a single high-resolution image. This paper is interested in the super resolution of multi-focus low-resolution images. It is assumed that, the images are acquired by low precision optics with limited depth of focus and from different viewpoints. Hence, the acquired images will be unregistered and multi-focus low-resolution images. The proposed SR reconstruction technique depends on using regularization-based schemes which have been demonstrated to be effective because SR reconstruction is actually an ill-posed problem. The proposed technique is based on using a local adaptive regularization parameter. This parameter can deal with the partial smoothness, which is located in images due to the multi-focus phenomenon. Moreover, the selection of the optimal value of this parameter is proposed using the particle swarm optimization method. The experimental results proved that the proposed SR reconstruction technique achieves better results than the more recent SR reconstruction technique named by locally adaptive bilateral total variation method which is also interested in treating the partial smoothness in low-resolution images by means of using a local adaptive regularization parameter.

Regional Spatially Adaptive Total Variation Super-Resolution With Spatial Information Filtering and Clustering

Total variation is used as a popular and effective image prior model in the regularization-based image processing fields. However, as the total variation model favors a piecewise constant solution, the processing result under high noise intensity in the flat regions of the image is often poor, and some pseudoedges are produced. In this paper, we develop a regional spatially adaptive total variation model. Initially, the spatial information is extracted based on each pixel, and then two filtering processes are added to suppress the effect of pseudoedges. In addition, the spatial information weight is constructed and classified with k-means clustering, and the regularization strength in each region is controlled by the clustering center value. The experimental results, on both simulated and real datasets, show that the proposed approach can effectively reduce the pseudoedges of the total variation regularization in the flat regions, and maintain the partial smoothness of the high-resolution image. More importantly, compared with the traditional pixel-based spatial information adaptive approach, the proposed region-based spatial information adaptive total variation model can better avoid the effect of noise on the spatial information extraction, and maintains robustness with changes in the noise intensity in the super-resolution process.

Partial Smoothness, Tilt Stability, and Generalized Hessians

We compare two recent variational-analytic approaches to second-order conditions and sensitivity analysis for nonsmooth optimization. We describe a broad setting where computing the generalized Hessian of Mordukhovich is easy. In this setting, the idea of tilt stability introduced by Poliquin and Rockafellar is equivalent to a classical smooth second-order condition.

The Binet–Legendre Metric in Finsler Geometry

For every Finsler metric $F$ we associate a Riemannian metric $g_F$ (called the Binet-Legendre metric). The transformation $F \mapsto g_F$ is $C^0$-stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric $g_F$ also behaves nicely under conformal or bilipshitz deformation of the Finsler metric $F$. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, solve a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend the classic result of H.C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Most proofs in this paper go along the following scheme: using the correspondence $F \mapsto g_F$we reduce the Finslerian problem to a similar problem for the Binet-Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem. Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by that of partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

Using partial smoothness of 𝑝-1 for factoring polynomials modulo 𝑝

Let an arbitrarily small positive constant δ \delta less than 1 1 and a polynomial f f with integer coefficients be fixed. We prove unconditionally that f f modulo p p can be completely factored in deterministic polynomial time if p − 1 p-1 has a ( ln ⁡ p ) O ( 1 ) (\ln p)^{O(1)} -smooth divisor exceeding p δ p^\delta . We also address the issue of factoring f f modulo p p over finite extensions of the prime field F p \mathbb {F}_p and show that p − 1 p-1 can be replaced by p k − 1 p^k-1 ( k ∈ N k\in \mathbb {N} ) for explicit classes of primes p p .

Nonsmooth optimization and robust control

Many questions of robust control analysis and synthesis fundamentally involve nonsmooth sets and functions, and their variational properties. Central examples include distances to instability and uncontrollability, the H ∞ norm, and pseudospectra. This plenary presentation at the Fifth IFAC Symposium on Robust Control Design (2006), surveys what current ideas from nonsmooth analysis say about the structure and conditioning of such functions and sets, and their numerical optimization. The presentation focuses on notions of nonsmooth derivatives and regularity and on structural tools such as partial smoothness and semi-algebraic techniques, illustrating how each idea helps in analysis and algorithm design. This survey relies heavily on joint work of the author with J.V. Burke (University of Washington) and M.L. Overton (Courant Institute).