A superquadratic infeasible-interior-point method for linear complementarity problems

We consider a modification of a path-following infeasible-interior- point algorithm described by Wright. In the new algorithm, we attempt to improve each new iterate by reusing the coefficient matrix factors from the latest step. We show that the modified algorithm has similar theoretical global convergence properties to the earlier algorithm, while its asymptotic convergence rate can be made superquadratic by an appropriate parameter choice

An interior point-proximal method of multipliers for linear positive semi-definite programming

In this paper we generalize the Interior Point-Proximal Method of Multipliers (IP-PMM) presented in Pougkakiotis and Gondzio (Comput Optim Appl 78:307–351, 2021. https://doi.org/10.1007/s10589-020-00240-9) for the solution of linear positive Semi-Definite Programming (SDP) problems, allowing inexactness in the solution of the associated Newton systems. In particular, we combine an infeasible Interior Point Method (IPM) with the Proximal Method of Multipliers (PMM) and interpret the algorithm (IP-PMM) as a primal-dual regularized IPM, suitable for solving SDP problems. We apply some iterations of an IPM to each sub-problem of the PMM until a satisfactory solution is found. We then update the PMM parameters, form a new IPM neighbourhood, and repeat this process. Given this framework, we prove polynomial complexity of the algorithm, under mild assumptions, and without requiring exact computations for the Newton directions. We furthermore provide a necessary condition for lack of strong duality, which can be used as a basis for constructing detection mechanisms for identifying pathological cases within IP-PMM.</p

Basis preconditioning in interior point methods

Solving normal equations AAᵀx = b, where A is an m x n matrix, is a common task in numerical optimization. For the efficient use of iterative methods, this thesis studies the class of preconditioners of the form BBᵀ , where B is a nonsingular "basis" matrix composed of m columns of A. It is known that for any matrix A of full row rank B can be chosen so that the entries in [B⁻¹A] are bounded by 1. Such a basis is said to have "maximum volume" and its preconditioner bounds the spectrum of the transformed normal matrix in the interval [1, 1+mn]. The theory is extended to (numerically) rank deficient matrices, yielding a rank revealing variant of Gaussian elimination and a method for computing the minimum norm solution for x from a reduced normal system and a low-rank update. Algorithms for finding a maximum volume basis are discussed. In the linear programming interior point method a sequence of normal equations needs to be solved, in which A changes by a column scaling from one system to the next. A heuristical algorithm is proposed for maintaining a basis of approximate maximum volume by update operations as those in the revised simplex method. Empirical results demonstrate that the approximation means no loss in the effectiveness of the preconditioner, but makes basis selection much more efficient. The implementation of an interior point solver based on the new linear algebra is described. Features of the code include the elimination of free variables during preconditioning and the removal of degenerate variables from the optimization process once sufficiently close to a bound. A crossover method recovers a vertex solution to the linear program, starting from the basis at the end of the interior point solve. A computational study shows that the implementation is robust and of general applicability, and that its average performance is comparable to that of state-of-the-art solvers

Final Iterations in Interior Point Models -- Preconditioned Conjugate Gradients and Modified Search Directions

In this article we consider modified search directions in the endgame of interior point methods for linear programming. In this stage, the normal equations determining the search directions become ill-conditioned. The modified search directions are computered by solving perturbed systems in which the systems may be solved efficiently by the preconditioned conjugate gradient solver. We prove the convergence of the interior point methods using the modified search directions and show that each barrier problem is solved with a superlinear convergence rate. A variation of Cholesky factorization is presented for computing a better preconditioner when the normal equations are ill-conditioned. These ideas have been implemented successfully and the numerical results show that the algorithms enhance the performance of the preconditioned conjugate gradients-based interior point methods

Solving Large-Scale AC Optimal Power Flow Problems Including Energy Storage, Renewable Generation, and Forecast Uncertainty

Renewable generation and energy storage are playing an ever increasing role in power systems. Hence, there is a growing need for integrating these resources into the optimal power flow (OPF) problem. While storage devices are important for mitigating renewable variability, they introduce temporal coupling in the OPF constraints, resulting in a multiperiod OPF formulation. This work explores a solution method for multiperiod AC OPF problems that combines a successive quadratic programming approach (AC-QP) with a second-order cone programming (SOCP) relaxation of the OPF problem. The solution of the SOCP relaxation is used to initialize the AC-QP OPF algorithm. Additionally, the lower bound on the objective value obtained from the SOCP relaxation provides a measure of solution quality. Compared to other initialization schemes, the SOCP-based approach offers improved convergence rate, execution time and solution quality. A reformulation of the the AC-QP OPF method that includes wind generation uncertainty is then presented. The resulting stochastic optimization problem is solved using a scenario based algorithm that is based on randomized methods that provide probabilistic guarantees of the solution. This approach produces an AC-feasible solution while satisfying reasonable reliability criteria. The proposed algorithm improves on techniques in prior work, as it does not rely upon model approximations and maintains scalability with respect to the number of scenarios considered in the OPF problem. The optimality of the proposed method is assessed using the lower bound from the solution of an SOCP relaxation and is shown to be sufficiently close to the globally optimal solution. Moreover, the reliability of the OPF solution is validated via Monte Carlo simulation and is demonstrated to fall within acceptable violation levels. Timing results are provided to emphasize the scalability of the method with respect to the number of scenarios considered and demonstrate its utility for real-time applications. Several extensions of this stochastic OPF are then developed for both operational and planning purposes. The first is to include the cost of generator reserve capacity in the objective of the stochastic OPF problem. The need for the increased accuracy provided by the AC OPF is highlighted by a case study that compares the reliability levels achieved by the AC-QP algorithm to those from the solution of a stochastic DC OPF. Next, the problem is extended to a planning context, determining the maximum wind penetration that can be added in a network while maintaining acceptable reliability criteria. The scalability of this planning method with respect not only to large numbers of wind scenarios but also to moderate network size is demonstrated. Finally, a formulation that minimizes both the cost of generation and the cost of reserve capacity while maximizing the wind generation added in the network is investigated. The proposed framework is then used to explore the inherent tradeoff between these competing objectives. A sensitivity study is then conducted to explore how the cost placed on generator reserve capacity can significantly impact the maximum wind penetration that can be reliably added in a network.PHDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138664/1/jkfelder_1.pd

Phylogeography and molecular systematics of species complexes in the genus Genetta (Carnivora, Viverridae)

The main aim of this study was to estimate phylogeographic patterns from mitochondrial DNA diversity and relate them with evolutionary structure in two species complexes of genets, Genetta genetta and Genetta "rubiginosa",which have fluid morphological variation. Both are widely distributed in sub-Saharan Africa but whereas G. "rubiginosa" appears in both closed and open habitats, G. genetta is absent from the rainforest and occurs also in the Maghreb, southwest Europe, and Arabia. DNA sequence data, mainly acquired from museum skin samples (about 75% of the total number of samples), was analysed using methods from the fields of phylogenetics and population genetics. The results for G. genetta are compatible with a scenario of allopatric fragmentation in grassland refuges during the Pleistocene climatic cycles as the main factor responsible for geographic genetic structure within Africa. The Arabian isolate showed significant genetic divergence, species-level compatible, and is probably the result of a long-distance dispersal from North Africa. Genetic diversity in Europe is a subset of that found in North Africa and shallow genetic distance is concordant with their anthropogenic introduction into Europe. North Africa seems to be cyclically connected to West and Central Africa during interglacial periods in which the Sahara recedes substantially. For G. "rubiginosa", isolated biogeographic relicts in the eastern African coast, possibly unsuspected species, were uncovered. However, the dominant pattern in the evolution of this species complex seems to be ecological differentiation in parapatry after invasion of open habitats from the rainforest ancestral habitat. Three general conclusions may be extracted from this study. Firstly, the use of mitochondrial DNA is clearly informative, but both gene flow among parapatric populations and introgression are confounding factors. Secondly, the Pleistocene in Africa has had variable biogeographical and evolutionary consequences for different taxa, depending on their ecological breadth. Lastly, large-scale utilisation of museum samples in phylogeographies of cryptic taxa has been rarely attempted but should become a standard approach