Multilevel algorithms for the optimization of structured problems

Although large scale optimization problems are very difficult to solve in general, problems that arise from practical applications often exhibit particular structure. In this thesis we study and improve algorithms that can efficiently solve structured problems. Three separate settings are considered. The first part concerns the topic of singularly perturbed Markov decision processes (MDPs). When a MDP is singularly perturbed, one can construct an aggregate model in which the solution is asymptotically optimal. We develop an algorithm that takes advantage of existing results to compute the solution of the original model. The proposed algorithm can compute the optimal solution with a reduction in complexity without any penalty in accuracy. In the second part, the class of empirical risk minimization (ERM) problems is studied. When using a first order method, the Lipschitz constant of the empirical risk plays a crucial role in the convergence analysis and stepsize strategy of these problems. We derive the probabilistic bounds for such Lipschitz constants using random matrix theory. Our results are used to derive the probabilistic complexity and develop a new stepsize strategy for first order methods. The proposed stepsize strategy, Probabilistic Upper-bound Guided stepsize strategy (PUG), has a strong theoretical guarantee on its performance compared to the standard stepsize strategy. In the third part, we extend the existing results on multilevel methods for unconstrained convex optimization. We study a special case where the hierarchy of models is created by approximating first and second order information of the exact model. This is known as Galerkin approximation, and we named the corresponding algorithm Galerkin-based Algebraic Multilevel Algorithm (GAMA). Three case studies are conducted to show how the structure of a problem could affect the convergence of GAMA.Open Acces

Computer Aided Verification

This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book

Formal Verification with Confidence Intervals to Establish Quality of Service Properties of Software Systems

Formal verification is used to establish the compliance of software and hardware systems with important classes of requirements. System compliance with functional requirements is frequently analyzed using techniques such as model checking, and theorem proving. In addition, a technique called quantitative verification supports the analysis of the reliability, performance, and other quality-of-service (QoS) properties of systems that exhibit stochastic behavior. In this paper, we extend the applicability of quantitative verification to the common scenario when the probabilities of transition between some or all states of the Markov models analyzed by the technique are unknown, but observations of these transitions are available. To this end, we introduce a theoretical framework, and a tool chain that establish confidence intervals for the QoS properties of a software system modelled as a Markov chain with uncertain transition probabilities. We use two case studies from different application domains to assess the effectiveness of the new quantitative verification technique. Our experiments show that disregarding the above source of uncertainty may significantly affect the accuracy of the verification results, leading to wrong decisions, and low-quality software systems

Generalized Sampling-Based Feedback Motion Planners

The motion planning problem can be formulated as a Markov decision process (MDP), if the uncertainties in the robot motion and environments can be modeled probabilistically. The complexity of solving these MDPs grow exponentially as the dimension of the problem increases and hence, it is nearly impossible to solve the problem even without constraints. Using hierarchical methods, these MDPs can be transformed into a semi-Markov decision process (SMDP) which only needs to be solved at certain landmark states. In the deterministic robotics motion planning community, sampling based algorithms like probabilistic roadmaps (PRM) and rapidly exploring random trees (RRTs) have been successful in solving very high dimensional deterministic problem. However they are not robust to system with uncertainties in the system dynamics and hence, one of the primary objective of this work is to generalize PRM/RRT to solve motion planning with uncertainty. We first present generalizations of randomized sampling based algorithms PRM and RRT, to incorporate the process uncertainty, and obstacle location uncertainty, termed as "generalized PRM" (GPRM) and "generalized RRT" (GRRT). The controllers used at the lower level of these planners are feedback controllers which ensure convergence of trajectories while mitigating the effects of process uncertainty. The results indicate that the algorithms solve the motion planning problem for a single agent in continuous state/control spaces in the presence of process uncertainty, and constraints such as obstacles and other state/input constraints. Secondly, a novel adaptive sampling technique, termed as "adaptive GPRM" (AGPRM), is proposed for these generalized planners to increase the efficiency and overall success probability of these planners. It was implemented on high-dimensional robot n-link manipulators, with up to 8 links, i.e. in a 16-dimensional state-space. The results demonstrate the ability of the proposed algorithm to handle the motion planning problem for highly non-linear systems in very high-dimensional state space. Finally, a solution methodology, termed the "multi-agent AGPRM" (MAGPRM), is proposed to solve the multi-agent motion planning problem under uncertainty. The technique uses a existing solution technique to the multiple traveling salesman problem (MTSP) in conjunction with GPRM. For real-time implementation, an ?inter-agent collision detection and avoidance? module was designed which ensures that no two agents collide at any time-step. Algorithm was tested on teams of homogeneous and heterogeneous agents in cluttered obstacle space and the algorithm demonstrate the ability to handle such problems in continuous state/control spaces in presence of process uncertainty

Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Computer Aided Verification

This open access two-volume set LNCS 10980 and 10981 constitutes the refereed proceedings of the 30th International Conference on Computer Aided Verification, CAV 2018, held in Oxford, UK, in July 2018. The 52 full and 13 tool papers presented together with 3 invited papers and 2 tutorials were carefully reviewed and selected from 215 submissions. The papers cover a wide range of topics and techniques, from algorithmic and logical foundations of verification to practical applications in distributed, networked, cyber-physical, and autonomous systems. They are organized in topical sections on model checking, program analysis using polyhedra, synthesis, learning, runtime verification, hybrid and timed systems, tools, probabilistic systems, static analysis, theory and security, SAT, SMT and decisions procedures, concurrency, and CPS, hardware, industrial applications