2,221 research outputs found
Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Closed-Loop Solvability
An optimal control problem is studied for a linear mean-field stochastic
differential equation with a quadratic cost functional. The coefficients and
the weighting matrices in the cost functional are all assumed to be
deterministic. Closed-loop strategies are introduced, which require to be
independent of initial states; and such a nature makes it very useful and
convenient in applications. In this paper, the existence of an optimal
closed-loop strategy for the system (also called the closed-loop solvability of
the problem) is characterized by the existence of a regular solution to the
coupled two (generalized) Riccati equations, together with some constraints on
the adapted solution to a linear backward stochastic differential equation and
a linear terminal value problem of an ordinary differential equation.Comment: 23 page
Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems
This paper is concerned with a stochastic linear quadratic (LQ, for short)
optimal control problem. The notions of open-loop and closed-loop solvabilities
are introduced. A simple example shows that these two solvabilities are
different. Closed-loop solvability is established by means of solvability of
the corresponding Riccati equation, which is implied by the uniform convexity
of the quadratic cost functional. Conditions ensuring the convexity of the cost
functional are discussed, including the issue that how negative the control
weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is
characterized by the convergence of the solutions to a family of Riccati
equations. Then, a minimizing sequence, whose convergence is equivalent to the
open-loop solvability of the problem, is constructed. Finally, an illustrative
example is presented.Comment: 40 page
Twin Solutions of Even Order Boundary Value Problems for Ordinary Differential Equations and Finite Difference Equations
The Avery-Henderson fixed-point theorem is first applied to obtain the existence of at least two positive solutions for the boundary value problem
(-1)ny(2n) = f(y); n = 1; 2; 3 ... and t 2 [0; 1];
with boundary conditions
y(2k)(0) = 0
y(2k+1)(1) = 0 for k = 0; 1; 2 ... n - 1:
This theorem is subsequently used to obtain the existence of at least two positive solutions for the dynamic boundary value problem
(-1)n (2n)u(k)g(u(k)); n = 1; 2; 3 .... and k (0; ... N);
with boundary conditions
(2k)u(0) = 0
(2k+1)u(N + 1) = 0 for k = 0; 1; 2 ... n 1
On the Geometry of Cyclic Lattices
Cyclic lattices are sublattices of ZN that are preserved under the rotational shift operator. Cyclic lattices were introduced by D.~Micciancio and their properties were studied in the recent years by several authors due to their importance in cryptography. In particular, Peikert and Rosen showed that on cyclic lattices in prime dimensions, the shortest independent vectors problem SIVP reduces to the shortest vector problem SVP with a particularly small loss in approximation factor, as compared to general lattices. In this paper, we further investigate geometric properties of cyclic lattices. Our main result is a counting estimate for the number of well-rounded cyclic lattices, indicating that well-rounded lattices are more common among cyclic lattices than generically. We also show that SVP is equivalent to SIVP on a positive proportion of Minkowskian well-rounded cyclic lattices in every dimension. As an example, we demonstrate an explicit construction of a family of such lattices on which this equivalence holds. To conclude, we introduce a class of sublattices of ZN closed under the action of subgroups of the permutation group SN, which are a natural generalization of cyclic lattices, and show that our results extend to all such lattices closed under the action of any N-cycle
Learning Rich Geographical Representations: Predicting Colorectal Cancer Survival in the State of Iowa
Neural networks are capable of learning rich, nonlinear feature
representations shown to be beneficial in many predictive tasks. In this work,
we use these models to explore the use of geographical features in predicting
colorectal cancer survival curves for patients in the state of Iowa, spanning
the years 1989 to 2012. Specifically, we compare model performance using a
newly defined metric -- area between the curves (ABC) -- to assess (a) whether
survival curves can be reasonably predicted for colorectal cancer patients in
the state of Iowa, (b) whether geographical features improve predictive
performance, and (c) whether a simple binary representation or richer, spectral
clustering-based representation perform better. Our findings suggest that
survival curves can be reasonably estimated on average, with predictive
performance deviating at the five-year survival mark. We also find that
geographical features improve predictive performance, and that the best
performance is obtained using richer, spectral analysis-elicited features.Comment: 8 page
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