Optimal Communication Structures for Concurrent Computing

This research focuses on communicative solvers that run concurrently and exchange information to improve performance. This “team of solvers” enables individual algorithms to communicate information regarding their progress and intermediate solutions, and allows them to synchronize memory structures with more “successful” counterparts. The result is that fewer nodes spend computational resources on “struggling” processes. The research is focused on optimization of communication structures that maximize algorithmic efficiency using the theoretical framework of Markov chains. Existing research addressing communication between the cooperative solvers on parallel systems lacks generality: Most studies consider a limited number of communication topologies and strategies, while the evaluation of different configurations is mostly limited to empirical testing. Currently, there is no theoretical framework for tuning communication between cooperative solvers to match the underlying hardware and software. Our goal is to provide such functionality by mapping solvers’ dynamics to Markov processes, and formulating the automatic tuning of communication as a well-defined optimization problem with an objective to maximize solvers’ performance metrics

Parallel Computing Applications in Large-Scale Power System Operations

Electrical energy is the basic necessity for the economic development of human societies. In recent decades, the electricity industry is undergoing enormous changes, which have evolved into a large-scale and competitive industry. The integration of volatile renewable energy, and the emergence of transmission switching (TS) techniques bring great challenges to the existing power system operations problems, especially security-constrained unit commitment (SCUC) solution engines. In order to deal with the uncertainty of volatile renewable energy, scenario-based stochastic optimization approach has been widely employed to ensure the reliability and economic of power systems, in which each scenario would represent a possible system situation. Meanwhile, the emergence of TS techniques allows the system operators to change the topology of transmission systems in order to improve economic benefits by mitigating transmission congestion. However, with the introduction of extra scenarios and decision variables, the complexity of the SCUC model increases dramatically and more computational efforts are required, which might make the power system operation problems difficult to solve and even intractable. Therefore, an advanced solution technique is urgently needed to solve both stochastic SCUC problems and TS-based SCUC problems in an effective and fast way. In this dissertation, a decomposition framework is presented for the optimal operation of the large-scale power system, which decomposes the original large-size power system optimization problem into smaller-size and tractable subproblems, and solves these decomposed subproblems in a parallel manner with the help of high performance computing techniques. Numerical case studies on a modified I 118-bus system and a practical 1168-bus system demonstrate the effectiveness and efficiency of the proposed approach which will offer the power system a secure and economic operation under various uncertainties and contingencies

Towards stronger Lagrangean bounds for stable spanning trees

Given a graph G=(V,E) and a set C of unordered pairs of edges regarded as being in conflict, a stable spanning tree in G is a set of edges T inducing a spanning tree in G, such that for each {e_i, e_j} in C, at most one of the edges e_i and e_j is in T. The existing work on Lagrangean algorithms to the NP-hard problem of finding minimum weight stable spanning trees is limited to relaxations with the integrality property. We have recently initiated the combinatorial and polyhedral study of fixed cardinality stable sets [see https://doi.org/10.1016/j.dam.2021.01. 019], which motivates a new formulation for stable spanning trees based on Lagrangean Decomposition. By optimizing over the spanning tree polytope of G and the fixed cardinality stable set polytope of the conflict graph H=(E,C) in the subproblems, we are able to determine stronger Lagrangean bounds (equivalent to dualizing exponentially-many subtour elimination constraints), while limiting the number of multipliers in the dual problem to |E|. This naturally asks for more sophisticated dual algorithms, requiring the fewest iterations possible, and we derive a collection of Lagrangean dual ascent directions to this end.publishedVersio

Enumerating a subset of the integer points inside a Minkowski sum

AbstractSparse elimination exploits the structure of algebraic equations in order to obtain tighter bounds on the number of roots and better complexity in numerically approximating them. The model of sparsity is of combinatorial nature, thus leading to certain problems in general-dimensional convex geometry. This work addresses one such problem, namely the computation of a certain subset of integer points in the interior of integer convex polytopes. These polytopes are Minkowski sums, but avoiding their explicit construction is precisely one of the main features of the algorithm. Complexity bounds for our algorithm are derived under certain hypotheses, in terms of output-size and the sparsity parameters. A public domain implementation is described and its performance studied. Linear optimization lies at the inner loop of the algorithm, hence we analyze the structure of the linear programs and compare different implementations

OSQP: An Operator Splitting Solver for Quadratic Programs

We present a general-purpose solver for convex quadratic programs based on the alternating direction method of multipliers, employing a novel operator splitting technique that requires the solution of a quasi-definite linear system with the same coefficient matrix at almost every iteration. Our algorithm is very robust, placing no requirements on the problem data such as positive definiteness of the objective function or linear independence of the constraint functions. It can be configured to be division-free once an initial matrix factorization is carried out, making it suitable for real-time applications in embedded systems. In addition, our technique is the first operator splitting method for quadratic programs able to reliably detect primal and dual infeasible problems from the algorithm iterates. The method also supports factorization caching and warm starting, making it particularly efficient when solving parametrized problems arising in finance, control, and machine learning. Our open-source C implementation OSQP has a small footprint, is library-free, and has been extensively tested on many problem instances from a wide variety of application areas. It is typically ten times faster than competing interior-point methods, and sometimes much more when factorization caching or warm start is used. OSQP has already shown a large impact with tens of thousands of users both in academia and in large corporations

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Decomposition and duality based approaches to stochastic integer programming

Stochastic Integer Programming is a variant of Linear Programming which incorporates integer and stochastic properties (i.e. some variables are discrete, and some properties of the problem are randomly determined after the first-stage decision). A Stochastic Integer Program may be rewritten as an equivalent Integer Program with a characteristic structure, but is often too large to effectively solve directly. In this thesis we develop new algorithms which exploit convex duality and scenario-wise decomposition of the equivalent Integer Program to find better dual bounds and faster optimal solutions. A major attraction of this approach is that these algorithms will be amenable to parallel computation