Related papers: Kazhdan-Lusztig Basis and Optimization

We present a subgradient method for minimizing non-smooth, non-Lipschitz convex optimization problems. The only structure assumed is that a strictly feasible point is known. We extend the work of Renegar [5] by taking a different…

We study representations of simply-laced Weyl groups which are equipped with canonical bases. Our main result is that for a large class of representations, the separable elements of the Weyl group $W$ act on these canonical bases by…

We describe an algorithm for computing the $p$-canonical basis of the Hecke algebra, or one of its antispherical modules. The algorithm does not operate in the Hecke category directly, but rather uses a faithful embedding of the Hecke…

We propose a new variant of Kelley's cutting-plane method for minimizing a nonsmooth convex Lipschitz-continuous function over the Euclidean space. We derive the method through a constructive approach and prove that it attains the optimal…

We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials…

This paper investigates the minimization of the expectation of piecewise polynomial loss functions over Wasserstein balls. This optimization problem often appears as a key sub-problem of distributionally robust optimization problems. We…

We consider the minimization of a continuous function over the intersection of a regular cone with an affine set via a new class of adaptive first- and second-order optimization methods, building on the Hessian-barrier techniques introduced…

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of [D. Borthwick, T. Paul and A. Uribe, Legendrian distributions with…

We give a simple and natural method for computing approximately optimal solutions for minimizing a convex function $f$ over a convex set $K$ given by a separation oracle. Our method utilizes the Frank--Wolfe algorithm over the cone of valid…

Parallel to the very rich theory of Kazhdan-Lusztig cells in characteristic $0$, we try to build a similar theory in positive characteristic. We study cells with respect to the $p$-canonical basis of the Hecke algebra of a crystallographic…

This paper is devoted to the theoretical and numerical investigation of an augmented Lagrangian method for the solution of optimization problems with geometric constraints. Specifically, we study situations where parts of the constraints…

In this paper, we continue our previous work on the reduction of algebraic lattices over imaginary quadratic fields for the special case when the lattice is spanned over a two dimensional basis. In particular, we show that the…

The (Iwahori-)Hecke algebra in the title is a $q$-deformation $\sH$ of the group algebra of a finite Weyl group $W$. The algebra $\sH$ has a natural enlargement to an endomorphism algebra $\sA=\End_\sH(\sT)$ where $\sT$ is a $q$-permutation…

Let G be a symplectic or orthogonal complex Lie group with Lie algebra g. As a G-module, the decomposition of the symmetric algebra S(g) into its irreducible components can be explicitely obtained by using identities due to Littlewood. We…

We introduce graded Hecke algebras H based on a (possibly disconnected) complex reductive group G and a cuspidal local system L on a unipotent orbit of a Levi subgroup M of G. These generalize the graded Hecke algebras defined and…

Convex geometries form a subclass of closure systems with unique criticals, or $UC$-systems. We show that the $F$-basis introduced in [1] for $UC$-systems, becomes optimum in convex geometries, in two essential parts of the basis: right…

In this paper, we prove a decomposition formula for the Kazhdan-Lusztig basis of affine Hecke algebras of rank 2 with positive weight function. Then we discuss some applications of this kind of decomposition to Lusztig's conjectures P1-P15.

We introduce a primal-dual framework for solving linearly constrained nonconvex composite optimization problems. Our approach is based on a newly developed Lagrangian, which incorporates \emph{false penalty} and dual smoothing terms. This…

We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an…

Kazhdan-Lusztig ideals, a family of generalized determinantal ideals investigated in [Woo-Yong '08], provide an explicit choice of coordinates and equations encoding a neighbourhood of a torus-fixed point of a Schubert variety on a type A…