Search results for "Mathematics::Optimization and Control"
showing 10 items of 30 documents
Integration of multifunctions with closed convex values in arbitrary Banach spaces
2018
Integral properties of multifunctions with closed convex values are studied. In this more general framework not all the tools and the technique used for weakly compact convex valued multifunctions work. We pay particular attention to the "positive multifunctions". Among them an investigation of multifunctions determined by vector-valued functions is presented. Finally, decomposition results are obtained for scalarly and gauge-defined integrals of multifunctions and a full description of McShane integrability in terms of Henstock and Pettis integrability is given.
Noncoincidence of Approximate and Limiting Subdifferentials of Integral Functionals
2011
For a locally Lipschitz integral functional $I_f$ on $L^1(T,\mathbf{R}^n)$ associated with a measurable integrand f, the limiting subdifferential and the approximate subdifferential never coincide at a point $x_0$ where $f(t,\cdot)$ is not subdifferentially regular at $x_0(t)$ for a.e. $t\in T$. The coincidence of both subdifferentials occurs on a dense set of $L^1(T,\mathbf{R}^n)$ if and only if $f(t,\cdot)$ is convex for a.e. $t\in T$. Our results allow us to characterize Aubin's Lipschitz-like property as well as the convexity of multivalued mappings between $L^1$-spaces. New necessary optimality conditions for some Bolza problems are also obtained.
Limited memory bundle algorithm for inequality constrained nondifferentiable optimization
2007
Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of hundreds or thousands of variables with various constraints. In this paper, we describe a new efficient adaptive limited memory interior point bundle method for large, possible nonconvex, nonsmooth inequality constrained optimization. The method is a hybrid of the nonsmooth variable metric bundle method and the smooth limited memory variable metric method, and the constraint handling is based on the primal-dual feasible direction interior point approach. The preliminary numerical experiments to be presented confirm the effectiveness of the method.
Test problems for large-scale nonsmooth minimization
2007
Many practical optimization problems involve nonsmooth (that is, not necessarily differentiable) functions of hundreds or thousands of variables with various constraints. However, there exist only few large-scale academic test problems for nonsmooth case and there is no established practice for testing solvers for large-scale nonsmooth optimization. For this reason, we now collect the nonsmooth test problems used in our previous numerical experiments and also give some new problems. Namely, we give problems for unconstrained, bound constrained, and inequality constrained nonsmooth minimization.
Approximations and Metric Regularity in Mathematical Programming in Banach Space
1993
This paper establishes verifiable conditions ensuring the important notion of metric regularity for general nondifferentiable programming problems in Banach spaces. These conditions are used to obtain Lagrange-Kuhn-Tucker multipliers for minimization problems with infinitely many inequality and equality constraints.
Cluster analysis for portfolio optimization
2005
We consider the problem of the statistical uncertainty of the correlation matrix in the optimization of a financial portfolio. We show that the use of clustering algorithms can improve the reliability of the portfolio in terms of the ratio between predicted and realized risk. Bootstrap analysis indicates that this improvement is obtained in a wide range of the parameters N (number of assets) and T (investment horizon). The predicted and realized risk level and the relative portfolio composition of the selected portfolio for a given value of the portfolio return are also investigated for each considered filtering method.
On a class of compactly epi-Lipschitzian sets
2003
The paper is devoted to the study of the so-called compactly epi-Lipschitzian sets. These sets are needed for many aspects of generalized differentiation, particulary for necessary optimality conditions, stability of mathematical programming problems and calculus rules for subdifferentials and normal cones. We present general conditions under which sets defined by general constraints are compactly epi-Lipschitzian. This allows us to show how the compact epi-Lipschitzness properties behave under set intersections.
Decompositions of Weakly Compact Valued Integrable Multifunctions
2020
We give a short overview on the decomposition property for integrable multifunctions, i.e., when an &ldquo
Delay in claim settlement and ruin probability approximations
1995
We introduce a general risk model for portfolios with delayed claims which is a natural extension of the classical Poisson model. We investigate ruin problems for different premium principles and provide approximations for the ruin probability. We conclude with some specific models, for example, for IBNR portfolios and portfolios where the pay-off process depends on the claim size.
On Limiting Fréchet ε-Subdifferentials
1998
This paper presents an e-sub differential calculus for nonconvex and nonsmooth functions. We extend the previous work by Jofre et all to the case where the functions are lower semicontinuous instead of locally Lipschitz.