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We propose a general dual ascent framework for Lagrangean decomposition of combinatorial problems. Although methods of this type have shown their efficiency for a number of problems, so far there was no general algorithm applicable to…
Time evolution equations for dynamical systems can often be derived from generating functionals. Examples are Newton's equations of motion in classical dynamics which can be generated within the Lagrange or the Hamiltonian formalism. We…
Despite the non-convexity of most modern machine learning parameterizations, Lagrangian duality has become a popular tool for addressing constrained learning problems. We revisit Augmented Lagrangian methods, which aim to mitigate the…
This paper presents the Gaussian subordination framework to generate optimal one-sided approximations to multidimensional real-valued functions by functions of prescribed exponential type. Such extremal problems date back to the works of…
We propose a duality in the relative Langlands program. This duality pairs a Hamiltonian space for a group $G$ with a Hamiltonian space under its dual group $\check{G}$, and recovers at a numerical level the relationship between a period on…
We survey research that studies the connection between the computational complexity of optimization problems on the one hand, and the duality gap between the primal and dual optimization problems on the other. To our knowledge, this is the…
The structure functions of the Lagrangian gauge algebra are given explicitly in terms of the hamiltonian constraints and the first order Hamiltonian structure functions and their derivatives.
This paper provides a new duality between entropy functions and network codes. Given a function $g\geq 0$ defined on all proper subsets of $N$ random variables, we provide a construction for a network multicast problem which is solvable if…
This paper presents a canonical duality theory for solving a general nonconvex constrained optimization problem within a unified framework to cover Lagrange multiplier method and KKT theory. It is proved that if both target function and…
Filinski constructed a symmetric lambda-calculus consisting of expressions and continuations which are symmetric, and functions which have duality. In his calculus, functions can be encoded to expressions and continuations using primitive…
In this note, we present two general classes of integral inequalities motivated by their applications to infinite dimensional systems. The inequalities possess general structures in terms of weight functions and lower quadratic bounds. Many…
We prove the existence of gaps between all the different classes of matrix monotone functions defined on an interval, provided the interval is non trivial and different from the whole real line. We then show how matrix monotone functions…
We identify a class of continuous compactly supported functions for which the known part of the Gabor frame set can be extended. At least for functions with support on an interval of length two, the curve determining the set touches the…
One of the central concepts in modern theoretical physics, gauge symmetry, is typically realised by lifting a finite-dimensional global symmetry group of a given functional to an infinite-dimensional local one by extending the functional to…
This paper gives a definition of g-harmonic functions and shows the relation between the g-harmonic functions and g-martingales. It's direct to construct such relation under smooth case, but for continuous case we need the theory of…
We extend the classical construction of operator colligations and characteristic functions. Consider the group $G$ of finite block unitary matrices of size $\alpha+\infty+...+\infty$ ($k$ times). Consider the subgroup $K=U(\infty)$, which…
We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…
We present two new classes of orthogonal functions, log orthogonal functions (LOFs) and generalized log orthogonal functions (GLOFs), which are constructed by applying a $\log$ mapping to Laguerre polynomials. We develop basic approximation…
By discussing several examples, the theory of generalized functional models is shown to be very natural for modeling some situations of reasoning under uncertainty. A generalized functional model is a pair (f, P) where f is a function…
Gaussian processes (GPs) are the main surrogate functions used for sequential modelling such as Bayesian Optimization and Active Learning. Their drawbacks are poor scaling with data and the need to run an optimization loop when using a…