460 research outputs found
GMRES-Accelerated ADMM for Quadratic Objectives
We consider the sequence acceleration problem for the alternating direction
method-of-multipliers (ADMM) applied to a class of equality-constrained
problems with strongly convex quadratic objectives, which frequently arise as
the Newton subproblem of interior-point methods. Within this context, the ADMM
update equations are linear, the iterates are confined within a Krylov
subspace, and the General Minimum RESidual (GMRES) algorithm is optimal in its
ability to accelerate convergence. The basic ADMM method solves a
-conditioned problem in iterations. We give
theoretical justification and numerical evidence that the GMRES-accelerated
variant consistently solves the same problem in iterations
for an order-of-magnitude reduction in iterations, despite a worst-case bound
of iterations. The method is shown to be competitive against
standard preconditioned Krylov subspace methods for saddle-point problems. The
method is embedded within SeDuMi, a popular open-source solver for conic
optimization written in MATLAB, and used to solve many large-scale semidefinite
programs with error that decreases like , instead of ,
where is the iteration index.Comment: 31 pages, 7 figures. Accepted for publication in SIAM Journal on
Optimization (SIOPT
Accelerating MCMC via Parallel Predictive Prefetching
We present a general framework for accelerating a large class of widely used
Markov chain Monte Carlo (MCMC) algorithms. Our approach exploits fast,
iterative approximations to the target density to speculatively evaluate many
potential future steps of the chain in parallel. The approach can accelerate
computation of the target distribution of a Bayesian inference problem, without
compromising exactness, by exploiting subsets of data. It takes advantage of
whatever parallel resources are available, but produces results exactly
equivalent to standard serial execution. In the initial burn-in phase of chain
evaluation, it achieves speedup over serial evaluation that is close to linear
in the number of available cores
Outer-space branch-and-bound algorithm for generalized linear multiplicative programs
This paper introduces a new global optimization algorithm for solving the
generalized linear multiplicative problem (GLMP). The algorithm starts by
introducing new variables and applying a logarithmic transformation
to convert the problem into an equivalent problem (EP). By using the strong
duality of linear program, a new convex relaxation subproblem is formulated to
obtain the lower bounds for the optimal value of EP. This relaxation
subproblem, combined with a simplicial branching process, forms the foundation
of a simplicial branch-and-bound algorithm that can globally solve the problem.
The paper also includes an analysis of the theoretical convergence and
computational complexity of the algorithm. Additionally, numerical experiments
are conducted to demonstrate the effectiveness of the proposed algorithm in
various test instances
Application-Specific Number Representation
Reconfigurable devices, such as Field Programmable Gate Arrays (FPGAs), enable application-
specific number representations. Well-known number formats include fixed-point, floating-
point, logarithmic number system (LNS), and residue number system (RNS). Such different
number representations lead to different arithmetic designs and error behaviours, thus produc-
ing implementations with different performance, accuracy, and cost.
To investigate the design options in number representations, the first part of this thesis presents
a platform that enables automated exploration of the number representation design space. The
second part of the thesis shows case studies that optimise the designs for area, latency or
throughput from the perspective of number representations.
Automated design space exploration in the first part addresses the following two major issues:
² Automation requires arithmetic unit generation. This thesis provides optimised
arithmetic library generators for logarithmic and residue arithmetic units, which support
a wide range of bit widths and achieve significant improvement over previous designs.
² Generation of arithmetic units requires specifying the bit widths for each
variable. This thesis describes an automatic bit-width optimisation tool called R-Tool,
which combines dynamic and static analysis methods, and supports different number
systems (fixed-point, floating-point, and LNS numbers).
Putting it all together, the second part explores the effects of application-specific number
representation on practical benchmarks, such as radiative Monte Carlo simulation, and seismic
imaging computations. Experimental results show that customising the number representations
brings benefits to hardware implementations: by selecting a more appropriate number format,
we can reduce the area cost by up to 73.5% and improve the throughput by 14.2% to 34.1%; by
performing the bit-width optimisation, we can further reduce the area cost by 9.7% to 17.3%.
On the performance side, hardware implementations with customised number formats achieve
5 to potentially over 40 times speedup over software implementations
An efficient outer space branch-and-bound algorithm for globally minimizing linear multiplicative problems
We propose an efficient outer space branch-and-bound algorithm for minimizing linear multiplicative problems (LMP). First, by introducing auxiliary variables, LMP is transformed into an equivalent problem (ELMP), where the number of auxiliary variables is equal to the number of linear functions. Subsequently, based on the properties of exponential and logarithmic functions, further equivalent transformation of ELMP is performed. Next, a novel linear relaxation technique is used to obtain the linear relaxation problem, which provides a reliable lower bound for the global optimal value of LMP. Once more, branching operation takes place in the outer space of the linear function while embedding compression technique to remove infeasible regions to the maximum extent possible, which significantly reduces the computational cost. Therefore, an outer space branch-and-bound algorithm is proposed. In addition, we conduct convergence analysis and complexity proof for the algorithm. Finally, the computational performance of the algorithm is demonstrated based on the experimental results obtained by testing a series of problems
(Global) Optimization: Historical notes and recent developments
Recent developments in (Global) Optimization are surveyed in this paper. We collected and commented quite a large number of recent references which, in our opinion, well represent the vivacity, deepness, and width of scope of current computational approaches and theoretical results about nonconvex optimization problems. Before the presentation of the recent developments, which are subdivided into two parts related to heuristic and exact approaches, respectively, we briefly sketch the origin of the discipline and observe what, from the initial attempts, survived, what was not considered at all as well as a few approaches which have been recently rediscovered, mostly in connection with machine learning
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