Lagrangian Relaxation Techniques for Scalable Spatial Conservation Planning

We address the problem of spatial conservation planning in which the goal is to maximize the expected spread of cascades of an endangered species by strategically purchasing land parcels within a given budget. This problem can be solved by standard integer programming methods using the sample average approximation (SAA) scheme. Our main contribution lies in exploiting the separable structure present in this problem and using Lagrangian relaxation techniques to gain scalability over the flat representation. We also generalize the approach to allow the application of the SAA scheme to a range of stochastic optimization problems. Our iterative approach is highly efficient in terms of space requirements and it provides an upper bound over the optimal solution at each iteration. We apply our approach to the Red-cockaded Woodpecker conservation problem. The results show that it can find the optimal solution significantly faster—sometimes by an order-of-magnitude—than using the flat representation for a range of budget sizes.

Stochastic Network Design: Models and Scalable Algorithms

Many natural and social phenomena occur in networks. Examples include the spread of information, ideas, and opinions through a social network, the propagation of an infectious disease among people, and the spread of species within an interconnected habitat network. The ability to modify a phenomenon towards some desired outcomes has widely recognized benefits to our society and the economy. The outcome of a phenomenon is largely determined by the topology or properties of its underlying network. A decision maker can take management actions to modify a network and, therefore, change the outcome of the phenomenon. A management action is an activity that changes the topology or other properties of a network. For example, species that live in a small area may expand their population and gradually spread into an interconnected habitat network. However, human development of various structures such as highways and factories may destroy natural habitats or block paths connecting different habitat patches, which results in a population decline. To facilitate the dispersal of species and help the population recover, artificial corridors (e.g., a wildlife highway crossing) can be built to restore connectivity of isolated habitats, and conservation areas can be established to restore historical habitats of species, both of which are examples of management actions. The set of management actions that can be taken is restricted by a budget, so we must find cost-effective allocations of limited funding resources. In the thesis, the problem of finding the (nearly) optimal set of management actions is formulated as a discrete and stochastic optimization problem. Specifically, a general decision-making framework called stochastic network design is defined to model a broad range of similar real-world problems. The framework is defined upon a stochastic network, in which edges are either present or absent with certain probabilities. It defines several metrics to measure the outcome of the underlying phenomenon and a set of management actions that modify the network or its parameters in specific ways. The goal is to select a subset of management actions, subject to a budget constraint, to maximize a specified metric. The major contribution of the thesis is to develop scalable algorithms to find high- quality solutions for different problems within the framework. In general, these problems are NP-hard, and their objective functions are neither submodular nor super-modular. Existing algorithms, such as greedy algorithms and heuristic search algorithms, either lack theoretical guarantees or have limited scalability. In the thesis, fast approximate algorithms are developed under three different settings that are gradually more general. The most restricted setting is when a network is tree-structured. For this case, fully polynomial-time approximation schemes (FPTAS) are developed using dynamic programming algorithms and rounding techniques. A more general setting is when networks are general directed graphs. We use a sampling technique to convert the original stochastic optimization problem into a deterministic optimization problem and develop a primal-dual algorithm to solve it efficiently. In the previous two problem settings, the goal is to maximize connectivity of networks. In the most general setting, the goal is to maximize the number of nodes being connected and minimize the distance between these connected nodes. For example, we do not only want the species to reach a large number of habitat areas but also want them to be able to get there within a reasonable amount of time. The scalable algorithms for this setting combine a fast primal-dual algorithm and a sampling procedure. Three real-world problems from the areas of computational sustainability and emergency response are used to evaluate these algorithms. They are the barrier removal problem aimed to determine which instream barriers to remove to help fish access their historical habitats in a river network, the spatial conservation planning problem to determine which habitat units to set as conservation areas to encourage the dispersal of endangered species in a landscape, and the pre-disaster preparation problem aimed to minimize the disruption of emergency medical services by natural disasters. In these three problems, the developed algorithms are much more scalable than the existing state-of-the-arts and produce high-quality solutions

Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today's power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a 0.71% optimality gap in no more than 720 seconds for the most challenging power system instances in the literature

XOR-Sampling for Network Design with Correlated Stochastic Events

Many network optimization problems can be formulated as stochastic network design problems in which edges are present or absent stochastically. Furthermore, protective actions can guarantee that edges will remain present. We consider the problem of finding the optimal protection strategy under a budget limit in order to maximize some connectivity measurements of the network. Previous approaches rely on the assumption that edges are independent. In this paper, we consider a more realistic setting where multiple edges are not independent due to natural disasters or regional events that make the states of multiple edges stochastically correlated. We use Markov Random Fields to model the correlation and define a new stochastic network design framework. We provide a novel algorithm based on Sample Average Approximation (SAA) coupled with a Gibbs or XOR sampler. The experimental results on real road network data show that the policies produced by SAA with the XOR sampler have higher quality and lower variance compared to SAA with Gibbs sampler.Comment: In Proceedings of the Twenty-sixth International Joint Conference on Artificial Intelligence (IJCAI-17). The first two authors contribute equall

Scheduling Conservation Designs for Maximum Flexibility via Network Cascade Optimization

One approach to conserving endangered species is to purchase and protect a set of land parcels in a way that maximizes the expected future population spread. Unfortunately, an ideal set of parcels may have a cost that is beyond the immediate budget constraints and must thus be purchased incrementally. This raises the challenge of deciding how to schedule the parcel purchases in a way that maximizes the flexibility of budget usage while keeping population spread loss in control. In this paper, we introduce a formulation of this scheduling problem that does not rely on knowing the future budgets of an organization. In particular, we consider scheduling purchases in a way that achieves a population spread no less than desired but delays purchases as long as possible. Such schedules offer conservation planners maximum flexibility and use available budgets in the most efficient way. We develop the problem formally as a stochastic optimization problem over a network cascade model describing a commonly used model of population spread. Our solution approach is based on reducing the stochastic problem to a novel variant of the directed Steiner tree problem, which we call the set-weighted directed Steiner graph problem. We show that this problem is computationally hard, motivating the development of a primal-dual algorithm for the problem that computes both a feasible solution and a bound on the quality of an optimal solution. We evaluate the approach on both real and synthetic conservation data with a standard population spread model. The algorithm is shown to produce near optimal results and is much more scalable than more generic off-the-shelf optimizers. Finally, we evaluate a variant of the algorithm to explore the trade-offs between budget savings and population growth

Online Relocating and Matching of Ride-Hailing Services: A Model-Based Modular Approach

This study proposes an innovative model-based modular approach (MMA) to dynamically optimize order matching and vehicle relocation in a ride-hailing platform. MMA utilizes a two-layer and modular modeling structure. The upper layer determines the spatial transfer patterns of vehicle flow within the system to maximize the total revenue of the current and future stages. With the guidance provided by the upper layer, the lower layer performs rapid vehicle-to-order matching and vehicle relocation. MMA is interpretable, and equipped with the customized and polynomial-time algorithm, which, as an online order-matching and vehicle-relocation algorithm, can scale past thousands of vehicles. We theoretically prove that the proposed algorithm can achieve the global optimum in stylized networks, while the numerical experiments based on both the toy network and realistic dataset demonstrate that MMA is capable of achieving superior systematic performance compared to batch matching and reinforcement-learning based methods. Moreover, its modular and lightweight modeling structure further enables it to achieve a high level of robustness against demand variation while maintaining a relatively low computational cost