IPM based sparse LP solver on a heterogeneous processor

The primal-dual interior point method (PDIPM) has proven to be an efficient tool for power system optimisation problems. Its computational efficiency heavily relies on sparsity techniques. Hence, when optimisation problems cannot be formulated into sparse form, PDIPM then may not be the right choice for these problems, because the computational efficiency drops significantly in factorisation of a dense matrix. A nonsparse power-system optimisation problem containing either-or constraints, pumped hydrostorage (PHS) scheduling, is presented and a two-level predictor-corrector version of PDIPM (PCPDIPM) is proposed to cope with this nonsparse and NP hard problem. To overcome the difficulty associated with the dense matrix structure, a special data transformation is proposed. By further exploiting the dense matrix structure, the performance of PCPDIPM is not deteriorated by the nonsparse structure. On the contrary, the computational efficiency is dramatically improved due to exploiting this structure. Moreover, to solve the difficulty associated with either-or constraints. An effective two-level LP procedure is proposed. To illustrate the performance of the proposed methodology, numerical results are carried out on two test cases. These results show that the presented two-level PCPDIPM solves the PHS scheduling effectively.

An interior-point algorithm for linear optimization based on a new barrier function

Scaling of geographic space refers to the fact that for a large geographic area its small constituents or units are much more common than the large ones. This article develops a novel perspective to the scaling of geographic space using large street networks involving both cities and countryside. Given a street network of an entire country, we decompose the street network into individual blocks, each of which forms a minimum ring or cycle such as city blocks and field blocks. The block sizes demonstrate the scaling property, that is, far more small blocks than large ones. Interestingly, we find that the mean of all the block sizes can easily separate between small and large blocks –a high percentage (e.g., 90%) of smaller ones and a low percentage (e.g., 10%) of larger ones. Based on this regularity, termed as the head/tail division rule, we propose an approach to delineating city boundaries by grouping the smaller blocks. The extracted city sizes for the three largest European countries (France, Germany, and United Kingdom) exhibit power law distributions. We further define the concept of border number as a topological distance of a block far from the outmost border to map the center(s) of the country and the city. We draw an analogy between a country (or a city or geographic space in general) and a complex organism like the human body or the human brain to further elaborate on the power of this block perspective in reflecting the structure or patterns of geographic space.

This study aims to improve the computation of the search direction in the primal-dual Interior Point Method through preconditioned iterative methods. It is about a hybrid approach that combines the Controlled Cholesky Factorization preconditioner and the Splitting preconditioner. This approach has shown good results, however, in these preconditioners there are factors that reduce their efficiency, such as faults on the diagonal when performing the Cholesky factorization, as well as a demand for excessive memory, among others. Thus, some modifications are proposed in these preconditioners, as well as a new phase change, in order toimprove the performance of the hybrid preconditioner. In the Controlled Cholesky Factorization, the parameters that control the filling and the correction of the faults which occur on the diagonal are modified. It considers the relationship between the components from Controlled Cholesky Factorization obtained before and after the fault on the diagonal. In the Splitting preconditioner, in turn, a sparse base is constructed through an appropriate ordering of the columns from constrained matrix optimization problem. In addition, a theoretical result is presented, which shows that, with the proposed ordering, the condition number of the preconditioned Normal Equation matrix with the Splitting preconditioner is uniformly limited by an amount that depends only on the original data of the problem and not on the iteration of the Interior Point Method. Numerical experiments with large scale problems, corroborate the robustness and computational efficiency from this approach.

This paper proposes an efficient computational technique for the optimal control of linear discrete-time systems subject to bounded disturbances with mixed linear constraints on the states and inputs. The problem of computing an optimal state feedback control policy, given the current state, is non-convex. A recent breakthrough has been the application of robust optimization techniques to reparameterize this problem as a convex program. While the reparameterized problem is theoretically tractable, the number of variables is quadratic in the number of stages or horizon length N and has no apparent exploitable structure, leading to computational time of $${\mathcal{O}}(N^{6})$$ per iteration of an interior-point method. We focus on the case when the disturbance set is ∞-norm bounded or the linear map of a hypercube, and the cost function involves the minimization of a quadratic cost. Here we make use of state variables to regain a sparse problem structure that is related to the structure of the original problem, that is, the policy optimization problem may be decomposed into a set of coupled finite horizon control problems. This decomposition can then be formulated as a highly structured quadratic program, solvable by primal-dual interior-point methods in which each iteration requires $${\mathcal{O}}(N^3)$$ time. This cubic iteration time can be guaranteed using a Riccati-based block factorization technique, which is standard in discrete-time optimal control. Numerical results are presented, using a standard sparse primal-dual interior point solver, that illustrate the efficiency of this approach.

Exploring complexity of large update interior-point methods for [formula omitted] linear complementarity problem based on Kernel function

This paper presents an implementation of the FDTD- compatible Green's function on a heterogeneous parallel processing system. The developed implementation simultaneously utilizes computational power of the central processing unit (CPU) and the graphics processing unit (GPU) to the computational tasks best suited for each architecture. Recently, closed-form expression for this discrete Green's function (DGF) was derived, which facilitates its applications in the FDTD simulations of radiation and scattering problems. Unfortunately, implementation of the new DGF formula in software requires a multiple precision arithmetic and may cause long runtimes. Therefore, an acceleration of the DGF computations on a CPU-GPU heterogeneous parallel processing system was developed using the multiple precision arithmetic and the OpenMP and CUDA parallel programming interfaces. The method avoids drawbacks of the CPU- and GPU-only accelerated implementations of the DGF, i.e., long runtime on the CPU and signiflcant overhead of the GPU initialization respectively for long and short length of the DGF waveform. As a result, the sevenfold speedup was obtained relative to the reference DGF implementation on a multicore CPU thus applicability of the DGF in FDTD simulations was signiflcantly improved.

The purpose of this paper is to obtain new complexity results for a second-order cone optimization (SOCO) problem. We define a proximity function for the SOCO by a kernel function. Furthermore we formulate an algorithm for a large-update primal-dual interior-point method (IPM) for the SOCO by using the proximity function and give its complexity analysis, and then we show that the new worst-case iteration bound for the IPM is O(q√N(logN)q+1/q log N/e), where q?1.

As indicated in the recent studies about primal-dual interior-point methods (IPMs) based on kernel functions, a kernel function not only serves to determine the search direction and measure the distance of the current iteration point to the $$\mu $$ -center, but also affects the iteration complexity and the practical computational efficiency of the algorithm. In this paper, we propose a new IPM for semidefinite optimization (SDO) based on a parameterized kernel function which is a generalization of the one presented by Bai et al. (Optim Methods Softw 17(6):985–1008, 2002). By using the good properties of the parameterized kernel function, we deduce that the iteration bound for large-update method is $$O(\sqrt{n}\log {n}\log {\frac{n}{\epsilon }})$$ for $$q=O(n)$$ , which is the best known complexity results for such methods. In our knowledge, this result is the first instance of primal-dual interior point method for SDO which involving the kernel function. Some numerical results have been provided.

We are concerned with structural optimization problems in CFD where the state variables are supposed to satisfy a linear or nonlinear Stokes system and the design variables are subject to bilateral pointwise constraints. Within a primal-dual setting, we suggest an all-at-once approach based on interior-point methods. The discretization is taken care of by Taylor-Hood elements with respect to a simplicial triangulation of the computational domain. The efficient numerical solution of the discretized problem relies on adaptive path-following techniques featuring a predictor-corrector scheme with inexact Newton solves of the KKT system by means of an iterative null-space approach. The performance of the suggested method is documented by several illustrative numerical examples.

We propose a modification that allows reducing the number of restarts in the computation of the Controlled Cholesky Factorization preconditioner. We use this preconditioner in the solution of linear systems arising from primal-dual interior point method. The Controlled Cholesky Factorization preconditioner depends on the fill-in parameter and the correction parameter that controls diagonal fault. We use geometric and algebraic tools to modify these parameters. In particular, we determine an equation whose exact solution avoids the diagonal fault. Numerical experiments with large-scale problems show that these modifications reduce the number of restarts. These experiments indicate that the new approach is robust and competitive.

The radioactive decay heat from nuclear waste packages may, depending on the thermal load, create coupled thermal-mechanical-hydrological-chemical (TMHC) processes in the near-field environment of a repository. A group of tests on a large block (LBT) are planned to provide a timely opportunity to test and calibrate some of the TMHC model concepts. The LBT is advantageous for testing and verifying model concepts because the boundary conditions are controlled, and the block can be characterized before and after the experiment. A block of Topopah Spring tuff of about 3 × 3 × 4.5 m was sawed and isolated at Fran Ridge, Nevada Test Site. Small blocks of the rock adjacent to the large block were collected for laboratory testing of some individual thermal-mechanical, hydrological, and chemical processes. A constant load of about 4 MPa will be applied to the top and sides of the large block. The sides will be sealed with moisture and thermal barriers. The large block will be heated by heaters within and guard heaters on the sides so that a dry-out zone and a condensate zone will exist simultaneously. Temperature, moisture content, pore pressure, chemical composition, stress, and displacement will be measured throughout the block during the heating and cool-down phases. The results from the experiments on small blocks and the tests on the large block will provide a better understanding of some concepts of the coupled TMHC processes. The progress of the project is presented in this paper.

Heterogeneous processors that mix big high performance cores with small low power cores promise excellent single-threaded performance coupled with high multi-threaded throughput and higher performance-per-watt. A significant portion of the commercial multicore heterogeneous processors are likely to have a common instruction set architecture( ISA). However, due to limited design resources and goals, each core is likely to contain ISA extensions not yet implemented in the other core. Therefore, such heterogeneous processors will have inherent functional asymmetry at the ISA level and face significant software challenges. This paper analyzes the software challenges to the operating system and the application layer software on a heterogeneous system with functional asymmetry, where the ISA of the small and big cores overlaps. We look at the widely deployed Intel® Architecture and propose solutions to the software challenges that arise when a heterogeneous processor is designed around it. We broadly categorize functional asymmetries into those that can be exposed to application software and those that should be handled by system software. While one can argue that new software written should be heterogeneity-aware, it is important that we find ways in which legacy software can extract the best performance from heterogeneous multicore systems.

In this paper we present a linear computational complexity framework for design and implementation of (constrained) lifted Iterative Learning Control (ILC) systems with quadratic cost. The problem of designing constrained lifted ILC with quadratic cost is formulated as a convex optimization problem. We solve this problem using the primal-dual interior point method. High computational complexity of the primal-dual method, which render this method computationally infeasible for high dimensional lifted ILC systems, is significantly decreased by exploiting the sequentially semi-separable (SSS) structure of lifted system matrices. More precisely, O(N <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) computational cost of one iteration of the primal-dual method is reduced to O(N), where N characterizes the size of the lifted system matrices. Furthermore, by exploiting the SSS structure the large lifted system matrices can be efficiently stored in computer memory. We also show that SSS structure can be exploited to efficiently implement analytical solution of the unconstrained lifted ILC problem with quadratic cost and for calculation of the norm and stability radius of ILC system.

An ε-sensitivity analysis in the primal–dual interior point method