An interior-point method for nonlinear programming is presented. It enjoys the flexibility of switching between a line search method that computes steps by factoring the primal-dual equations and a trust region method that uses a conjugate gradient iteration. Steps computed by direct factorization are always tried first, but if they are deemed ineffective, a trust region iteration that guarantees progress toward stationarity is invoked. To demonstrate its effectiveness, the algorithm is implemented in the Knitro [6, 28] software package and is extensively tested on a wide selection of test problems. 1

This paper describes Knitro 5.0, a C-package for nonlinear optimization that combines complementary approaches to nonlinear optimization to achieve robust performance over a wide range of application requirements. The package is designed for solving large-scale, smooth nonlinear programming problems, and it is also effective for the following special cases: unconstrained optimization, nonlinear systems of equations, least squares, and linear and quadratic programming. Various algorithmic options are available, including two interior methods and an active-set method. The package provides crossover techniques between algorithmic options as well as automatic selection of options and settings. 1

Network operators control the flow of traffic through their networks by adapting the configuration of the underlying routing protocols. For example, they tune the integer link weights that interior gateway protocols like OSPF and IS-IS use to compute shortest paths. The resulting optimization problem—to find the best link weights for a given topology and traffic matrix—is computationally intractable even for the simplest objective functions, forcing the use of local-search techniques. The optimization problem is difficult because these protocols split traffic evenly along shortest paths, with no ability to adjust the splitting percentages or direct traffic on other paths. In this paper, we propose an extension to these protocols, called Distributed Exponentially-weighted Flow SpliTting (DEFT), where the routers can direct traffic on non-shortest paths, with an exponential penalty on longer paths. DEFT leads not only to a simpler optimization problem, but also to weight settings that provably perform always better than OSPF and IS-IS. In the optimization problem we present, both link weights and flows of traffic are integrated as optimization variables into the formulation and jointly solved by a two-stage iterative method. Our novel formulation leads to a much more efficient way to identify good link weights than the local-search heuristics used for OSPF and IS-IS today. DEFT retains the simplicity of having routers compute paths based on configurable link weights, while approaching the performance of more complex routing protocols that can split traffic arbitrarily over any paths.

Processors progressively age during their service life due to normal workload activity. Such aging results in gradually slower circuits. Anticipating this fact, designers add timing guardbands to processors, so that processors last for a number of years. As a result, aging has important design and cost implications. To address this problem, this paper shows how to hide the effects of aging and how to slow it down. Our framework is called Facelift. It hides aging through aging-driven application scheduling. It slows down aging by applying voltage changes at key times — it uses a non-linear optimization algorithm to carefully balance the impact of voltage changes on the aging rate and on the critical path delays. Moreover, Facelift can gainfully configure the chip for a short service life. Simulation results indicate that Facelift leads to more cost-effective multicores. We can take a multicore designed for a 7-year service life and, by hiding and slowing down aging, enable it to run, on average, at a 14–15% higher frequency during its whole service life. Alternatively, we can design the multicore for a 5 to 7-month service life and still use it for 7 years. 1

Abstract. Maximum weight matchings have become an important tool for solving highly indefinite unsymmetric linear systems, especially in direct solvers. In this study we investigate the benefit of reorderings and scalings based on symmetrized maximum weight matchings as a preprocessing step for incomplete LDL T factorizations. The reorderings are constructed such that the matched entries form 1 × 1or2 × 2 diagonal blocks in order to increase the diagonal dominance of the system. During the incomplete factorization only tridiagonal pivoting is used. We report results for this approach and comparisons with other solution methods for a diverse set of symmetric indefinite matrices, ranging from nonlinear elasticity to interior point optimization.

This paper reviews, extends and analyzes a new class of penalty methods for nonlinear optimization. These methods adjust the penalty parameter dynamically; by controlling the degree of linear feasibility achieved at every iteration, they promote balanced progress toward optimality and feasibility. In contrast with classical approaches, the choice of the penalty parameter ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential quadratic programming (SQP) and sequential linear-quadratic programming (SLQP) methods that use trust regions to promote convergence. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.

Abstract This paper considers strategies for selecting the barrier parameter at every iterationof an interior-point method for nonlinear programming. Numerical experiments suggest that adaptive choices, such as Mehrotra's probing procedure, outperform static strate-gies that hold the barrier parameter fixed until a barrier optimality test is satisfied. A new adaptive strategy is proposed based on the minimization of a quality function. Thepaper also proposes a globalization framework that ensures the convergence of adaptive interior methods. The barrier update strategies proposed in this paper are applica-ble to a wide class of interior methods and are tested in the two distinct algorithmic frameworks provided by the ipopt and knitro software packages.

This paper reviews the development of exact penalty methods for nonlinear optimization and discusses their increasingly important role in optimization algorithms and software. In their most recent stage of development, penalty methods adjust the penalty parameter dynamically. By controlling the degree of linear feasibility achieved at every iteration, these methods balance progress toward optimality and feasibility. The choice of the penalty parameter thus ceases to be a heuristic and is determined, instead, by a subproblem with clearly defined objectives. The new penalty update strategy is presented in the context of sequential linear-quadratic penalty methods, and is then extended to sequential quadratic programming. The paper concludes with a discussion of penalty parameters for merit functions used in line search methods.

Abstract — We report on our experience with integrating and using graphics processing units (GPUs) as fast parallel floatingpoint co-processors to accelerate two fundamental computational scientific kernels on the GPU: sparse direct factorization and nonlinear interior-point optimization. Since a full re-implementation of these complex kernels is typically not feasible, we identify the matrix-matrix multiplication as a first natural entry-point for a minimally invasive integration of GPUs. We investigate the performance on the NVIDIA GeForce 8800 multicore chip initially architectured for intensive gaming applications. We exploit the architectural features of the GeForce 8800 GPU to design an efficient GPU-parallel sparse matrix solver. A prototype approach to leverage the bandwidth and computing power of GPUs for these matrix kernel operation is demonstrated resulting in an overall performance of over 110 GFlops/s on the desktop for large matrices. We use our GPU algorithm for PDE-constrained optimization problems and demonstrate that the commodity GPU is a useful co-processor for scientific applications. Index Terms — Parallel processing, graphics processing units, matrix decomposition, sparse direct solvers, nonlinear optimization I.

This paper discusses how to build a solver for mixed integer quadratically constrained programs (MIQCPs) by extending a framework for constraint integer programming (CIP). The advantage of this approach is that we can utilize the full power of advanced MILP and CP technologies, in particular for the linear relaxation and the discrete components of the problem. We use an outer approximation generated by linearization of convex constraints and linear underestimation of nonconvex constraints to relax the problem. Further, we give an overview of the reformulation, separation, and propagation techniques that are used to handle the quadratic constraints efficiently. We implemented these methods in the branch-cut-and-price framework SCIP. Computational experiments indicating the potential of the approach and evaluating the impact of the algorithmic components are provided.