374 research outputs found
Multilevel quasiseparable matrices in PDE-constrained optimization
Optimization problems with constraints in the form of a partial differential
equation arise frequently in the process of engineering design. The
discretization of PDE-constrained optimization problems results in large-scale
linear systems of saddle-point type. In this paper we propose and develop a
novel approach to solving such systems by exploiting so-called quasiseparable
matrices. One may think of a usual quasiseparable matrix as of a discrete
analog of the Green's function of a one-dimensional differential operator. Nice
feature of such matrices is that almost every algorithm which employs them has
linear complexity. We extend the application of quasiseparable matrices to
problems in higher dimensions. Namely, we construct a class of preconditioners
which can be computed and applied at a linear computational cost. Their use
with appropriate Krylov methods leads to algorithms of nearly linear
complexity
Convergence Analysis of an Inexact Feasible Interior Point Method for Convex Quadratic Programming
In this paper we will discuss two variants of an inexact feasible interior
point algorithm for convex quadratic programming. We will consider two
different neighbourhoods: a (small) one induced by the use of the Euclidean
norm which yields a short-step algorithm and a symmetric one induced by the use
of the infinity norm which yields a (practical) long-step algorithm. Both
algorithms allow for the Newton equation system to be solved inexactly. For
both algorithms we will provide conditions for the level of error acceptable in
the Newton equation and establish the worst-case complexity results
Solving Large-Scale Optimization Problems Related to Bell's Theorem
Impossibility of finding local realistic models for quantum correlations due
to entanglement is an important fact in foundations of quantum physics, gaining
now new applications in quantum information theory. We present an in-depth
description of a method of testing the existence of such models, which involves
two levels of optimization: a higher-level non-linear task and a lower-level
linear programming (LP) task. The article compares the performances of the
existing implementation of the method, where the LPs are solved with the
simplex method, and our new implementation, where the LPs are solved with a
matrix-free interior point method. We describe in detail how the latter can be
applied to our problem, discuss the basic scenario and possible improvements
and how they impact on overall performance. Significant performance advantage
of the matrix-free interior point method over the simplex method is confirmed
by extensive computational results. The new method is able to solve problems
which are orders of magnitude larger. Consequently, the noise resistance of the
non-classicality of correlations of several types of quantum states, which has
never been computed before, can now be efficiently determined. An extensive set
of data in the form of tables and graphics is presented and discussed. The
article is intended for all audiences, no quantum-mechanical background is
necessary.Comment: 19 pages, 7 tables, 1 figur
Using the primal-dual interior point algorithm within the branch-price-and-cut method
AbstractBranch-price-and-cut has proven to be a powerful method for solving integer programming problems. It combines decomposition techniques with the generation of both columns and valid inequalities and relies on strong bounds to guide the search in the branch-and-bound tree. In this paper, we present how to improve the performance of a branch-price-and-cut method by using the primal-dual interior point algorithm. We discuss in detail how to deal with the challenges of using the interior point algorithm with the core components of the branch-price-and-cut method. The effort to overcome the difficulties pays off in a number of advantageous features offered by the new approach. We present the computational results of solving well-known instances of the vehicle routing problem with time windows, a challenging integer programming problem. The results indicate that the proposed approach delivers the best overall performance when compared with a similar branch-price-and-cut method which is based on the simplex algorithm
Non-parametric liquidity-adjusted VaR model: a stochastic programming approach
This paper proposes a Stochastic Programming (SP) approach for the calculation of the liquidity-adjusted Value-at-Risk (LVaR). The model presented in this paper offers an alternative to Almgren and Chriss’s mean-variance approach (1999 and 2000). In this research, a two-stage stochastic programming model is developed with the intention of deriving the optimal trading strategies that respond dynamically to a given market situation. The sample paths approach is adopted for scenario generation. The scenarios are thus represented by a collection of simulated sample paths rather than the ‘tree structure’ usually employed in stochastic programming. Consequently, the SP LVaR presented in this paper can be considered as a non-parametric approach, which is in contrast to Almgren and Chriss’s parametric solution. Initially, a set of numerical experiments indicates that the LVaR figures are quite similar for both approaches when all the underlying financial assumptions are identical. Following this sanity check, a second set of numerical experiments shows how the randomness of the different types (i.e., Bid and Ask spread) can be easily incorporated into the problem due to the stochastic programming formulation and how optimal and adaptive trading strategies can be derived through a two-stage structure (i.e., a ‘recourse’ problem). Hence, the results presented in this paper allow the introduction of new dimensionalities into the computation of LVaR by incorporating different market conditions.Liquidity adjusted Value at Risk (LVaR); Liquidation Cost; Stochastic programming; Optimal trading strategy; Non-parametric approach; Sample paths.
Solving a Class of LP Problems with a Primal-Dual Logarithmic Barrier Method
Applying a higher order primal-dual logarithmic barrier method for solving large real-life linear programming problems is addressed in this paper. The efficiency of interior point algorithm on these problems is compared with the one of the state-of-the-art simplex code MINOS version 5.3. Based on such experience, a wide class of LP problems is identified for which logarithmic barrier approach seems advantageous over the simplex one. Additionally, some practical rules for model builders are derived that should allow them to create problems that can easily be solved with logarithmic barrier algorithms
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