221 research outputs found
Chance Constrained Mixed Integer Program: Bilinear and Linear Formulations, and Benders Decomposition
In this paper, we study chance constrained mixed integer program with
consideration of recourse decisions and their incurred cost, developed on a
finite discrete scenario set. Through studying a non-traditional bilinear mixed
integer formulation, we derive its linear counterparts and show that they could
be stronger than existing linear formulations. We also develop a variant of
Jensen's inequality that extends the one for stochastic program. To solve this
challenging problem, we present a variant of Benders decomposition method in
bilinear form, which actually provides an easy-to-use algorithm framework for
further improvements, along with a few enhancement strategies based on
structural properties or Jensen's inequality. Computational study shows that
the presented Benders decomposition method, jointly with appropriate
enhancement techniques, outperforms a commercial solver by an order of
magnitude on solving chance constrained program or detecting its infeasibility
Robust Combinatorial Optimization with Locally Budgeted Uncertainty
Budgeted uncertainty sets have been established as a major influence on
uncertainty modeling for robust optimization problems. A drawback of such sets
is that the budget constraint only restricts the global amount of cost increase
that can be distributed by an adversary. Local restrictions, while being
important for many applications, cannot be modeled this way.
We introduce new variant of budgeted uncertainty sets, called locally
budgeted uncertainty. In this setting, the uncertain parameters become
partitioned, such that a classic budgeted uncertainty set applies to each
partition, called region.
In a theoretical analysis, we show that the robust counterpart of such
problems for a constant number of regions remains solvable in polynomial time,
if the underlying nominal problem can be solved in polynomial time as well. If
the number of regions is unbounded, we show that the robust selection problem
remains solvable in polynomial time, while also providing hardness results for
other combinatorial problems.
In computational experiments using both random and real-world data, we show
that using locally budgeted uncertainty sets can have considerable advantages
over classic budgeted uncertainty sets
Essays on Entertainment Analytics
This thesis explores live entertainment analytics and revenue management allocation strategies for live entertainment.
In Chapter two, we look at empirical factors that effect the success of Broadway shows. How well-known actors (stars) effect film revenues has been a recurring question of entertainment producers and academics. Because a film cannot be disentangled from a star involved, researchers have long struggled to rule out ``reverse-causality\u27\u27 - that stars have access to higher quality movies. Using a novel data set that includes Broadway show revenues and actor usage, we provide a fixed-effects regression and case studies. We find across multiple specifications that increases in star power in a show improve revenue. Motivated by social grouping and the associated operational challenges, in Chapter three we formulate and study extensions to the Dynamic Stochastic Knapsack Problem (DSKP). We compartmentalize the knapsack according to predefined reward-to-weight ratios, and incorporate a stochastic interaction between the offered set of open compartments and the item placement. Using a specific interaction function inspired by customer choice in the entertainment industry, we provide an algorithm to determine the optimal solution and obtain insights into structural properties. Given the computational complexity of the dynamic program we also propose and analyze via simulation a heuristic algorithm. In Chapter four, in a large sequence of simulations, we propose and study practical heuristic algorithms on which seats should be offered to requests. We propose an algorithm that offers revenue improvements from a ``naive\u27\u27 policy on the order of 5-10%.
Throughout, we aim for managerial relevance, providing implications to current techniques both in strategy as well as operations
Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data
Constraint Programming (CP) has proved an effective paradigm to model and
solve difficult combinatorial satisfaction and optimisation problems from
disparate domains. Many such problems arising from the commercial world are
permeated by data uncertainty. Existing CP approaches that accommodate
uncertainty are less suited to uncertainty arising due to incomplete and
erroneous data, because they do not build reliable models and solutions
guaranteed to address the user's genuine problem as she perceives it. Other
fields such as reliable computation offer combinations of models and associated
methods to handle these types of uncertain data, but lack an expressive
framework characterising the resolution methodology independently of the model.
We present a unifying framework that extends the CP formalism in both model
and solutions, to tackle ill-defined combinatorial problems with incomplete or
erroneous data. The certainty closure framework brings together modelling and
solving methodologies from different fields into the CP paradigm to provide
reliable and efficient approches for uncertain constraint problems. We
demonstrate the applicability of the framework on a case study in network
diagnosis. We define resolution forms that give generic templates, and their
associated operational semantics, to derive practical solution methods for
reliable solutions.Comment: Revised versio
Multi-index Stochastic Collocation convergence rates for random PDEs with parametric regularity
We analyze the recent Multi-index Stochastic Collocation (MISC) method for
computing statistics of the solution of a partial differential equation (PDEs)
with random data, where the random coefficient is parametrized by means of a
countable sequence of terms in a suitable expansion. MISC is a combination
technique based on mixed differences of spatial approximations and quadratures
over the space of random data and, naturally, the error analysis uses the joint
regularity of the solution with respect to both the variables in the physical
domain and parametric variables. In MISC, the number of problem solutions
performed at each discretization level is not determined by balancing the
spatial and stochastic components of the error, but rather by suitably
extending the knapsack-problem approach employed in the construction of the
quasi-optimal sparse-grids and Multi-index Monte Carlo methods. We use a greedy
optimization procedure to select the most effective mixed differences to
include in the MISC estimator. We apply our theoretical estimates to a linear
elliptic PDEs in which the log-diffusion coefficient is modeled as a random
field, with a covariance similar to a Mat\'ern model, whose realizations have
spatial regularity determined by a scalar parameter. We conduct a complexity
analysis based on a summability argument showing algebraic rates of convergence
with respect to the overall computational work. The rate of convergence depends
on the smoothness parameter, the physical dimensionality and the efficiency of
the linear solver. Numerical experiments show the effectiveness of MISC in this
infinite-dimensional setting compared with the Multi-index Monte Carlo method
and compare the convergence rate against the rates predicted in our theoretical
analysis
Estimation and Control of Dynamical Systems with Applications to Multi-Processor Systems
System and control theory is playing an increasingly important role in the design and analysis of computing systems. This thesis investigates a set of estimation and control problems that are driven by new challenges presented by next-generation Multi-Processor Systems on Chips (MPSoCs). Specifically, we consider problems related to state norm estimation, state estimation for positive systems, sensor selection, and nonlinear output tracking. Although these problems are motivated by applications to multi-processor systems, the corresponding theory and algorithms are developed for general dynamical systems.
We first study state norm estimation for linear systems with unknown inputs. Specifically, we consider a formulation where the unknown inputs and initial condition of the system are bounded in magnitude, and the objective is to construct an unknown input norm-observer which estimates an upper bound for the norm of the states. This class of problems is motivated by the need to estimate the maximum temperature across a multi-core processor, based on a given model of the thermal dynamics. In order to characterize the existence of the norm observer, we propose a notion of bounded-input-bounded-output-bounded-state (BIBOBS) stability; this concept supplements various system properties, including bounded-input-bounded-output (BIBO) stability, bounded-input-bounded-state (BIBS) stability, and input-output-to-state stability (IOSS).We provide necessary and sufficient conditions on the system matrices under which a linear system is BIBOBS stable, and show that the set of modes of the system with magnitude 1 plays a key role. A construction for the unknown input norm-observer follows as a byproduct.
Then we investigate the state estimation problem for positive linear systems with unknown inputs. This problem is also motivated by the need to monitor the temperature of a multi-processor system and the property of positivity arises due to the physical nature of the thermal model. We extend the concept of strong observability to positive systems and as a negative result, we show that the additional information on positivity does not help in state estimation. Since the states
of the system are always positive, negative state estimates are meaningless and the positivity of the observers themselves may be desirable in certain applications. Moreover, positive systems possess certain desired robustness properties. Thus, for positive systems where state estimation with unknown inputs is possible, we provide a linear programming based design procedure for delayed positive observers.
Next we consider the problem of selecting an optimal set of sensors to estimate the states of linear dynamical systems; in the context of multi-core processors, this problem arises due to the need to place thermal sensors in order to perform state estimation. The goal is to choose (at design-time) a subset of sensors (satisfying certain budget constraints) from a given set in order to minimize the trace of the steady state a priori or a posteriori error covariance produced by a Kalman filter. We show that the a priori and a posteriori error covariance-based sensor selection problems are both NP-hard, even under the additional assumption that the system is stable. We then provide bounds on the worst-case performance of sensor selection algorithms based on the system dynamics, and show that certain greedy algorithms are optimal for two classes of systems. However, as a negative result, we show that certain typical objective functions are not submodular or supermodular in general. While this makes it difficult to evaluate the performance of greedy algorithms for sensor selection (outside of certain special cases), we show via simulations that these greedy algorithms perform well in practice.
Finally, we study the output tracking problem for nonlinear systems with constraints. This class of problems arises due to the need to optimize the energy consumption of the CPU-GPU subsystem in multi-processor systems while satisfying certain Quality of Service (QoS) requirements. In order for the system output to track a class of bounded reference signals with limited
online computational resources, we propose a sampling-based explicit nonlinear model predictive control (ENMPC) approach, where only a bound on the admissible references is known to the designer a priori. The basic idea of sampling-based ENMPC is to sample the state and reference signal space using deterministic sampling and construct the ENMPC by using regression
methods. The proposed approach guarantees feasibility and stability for all admissible references and ensures asymptotic convergence to the set-point. Furthermore, robustness through the use of an ancillary controller is added to the nominal ENMPC for a class of nonlinear systems with additive disturbances, where the robust controller keeps the system output close to the desired nominal trajectory
A branch-and-bound algorithm for stable scheduling in single-machine production systems.
Robust scheduling aims at the construction of a schedule that is protected against uncertain events. A stable schedule is a robust schedule that will change little when variations in the input parameters arise. This paper proposes a branch-and-bound algorithm for optimally solving a single-machine scheduling problem with stability objective, when a single job is anticipated to be disrupted.Branch-and-bound; Construction; Event; Job; Robust scheduling; Robustness; Scheduling; Single-machine scheduling; Stability; Systems; Uncertainty;
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