Solution Techniques for Classes of Biobjective and Parametric Programs

Mathematical optimization, or mathematical programming, has been studied for several decades. Researchers are constantly searching for optimization techniques which allow one to de-termine the ideal course of action in extremely complex situations. This line of scientific inquiry motivates the primary focus of this dissertation — nontraditional optimization problems having either multiple objective functions or parametric input. Utilizing multiple objective functions al-lows one to account for the fact that the decision process in many real-life problems in engineering, business, and management is often driven by several conflicting criteria such as cost, performance, reliability, safety, and productivity. Additionally, incorporating parametric input allows one to ac-count for uncertainty in models’ data, which can arise for a number of reasons, including a changing availability of resources, estimation or measurement errors, or implementation errors caused by stor-ing data in a fixed precision format. However, when a decision problem has either parametric input or multiple objectives, one cannot hope to find a single, satisfactory solution. Thus, in this work we develop techniques which can be used to determine sets of desirable solutions. The two main problems we consider in this work are the biobjective mixed integer linear program (BOMILP) and the multiparametric linear complementarity problem (mpLCP). BOMILPs are optimization problems in which two linear objectives are optimized over a polyhedron while restricting some of the decision variables to be integer. We present a new data structure in the form of a modified binary tree that can be used to store the solution set of BOMILP. Empirical evidence is provided showing that this structure is able to store these solution sets more efficiently than other data structures that are typically used for this purpose. We also develop a branch-and-bound (BB) procedure that can be used to compute the solution set of BOMILP. Computational experiments are conducted in order to compare the performance of the new BB procedure with current state-of-the-art methods for determining the solution set of BOMILP. The results provide strong evidence of the utility of the proposed BB method. We also present new procedures for solving two variants of the mpLCP. Each of these proce-dures consists of two phases. In the first phase an initial feasible solution to mpLCP which satisfies certain criteria is determined. This contribution alone is significant because the question of how such an initial solution could be generated was previously unanswered. In the second phase the set of fea-sible parameters is partitioned into regions such that the solution of the mpLCP, as a function of the parameters, is invariant over each region. For the first variant of mpLCP, the worst-case complex-ity of the presented procedure matches that of current state-of-the-art methods for nondegenerate problems and is lower than that of current state-of-the-art methods for degenerate problems. Addi-tionally, computational results show that the proposed procedure significantly outperforms current state-of-the-art methods in practice. The second variant of mpLCP we consider was previously un-solved. In order to develop a solution strategy, we first study the structure of the problem in detail. This study relies on the integration of several key concepts from algebraic geometry and topology into the field of operations research. Using these tools we build the theoretical foundation necessary to solve the mpLCP and propose a strategy for doing so. Experimental results indicate that the presented solution method also performs well in practice

Branch-and-bound for biobjective mixed-integer linear programming

We present a generic branch-and-bound method for finding all the Pareto solutions of a biobjective mixed integer program. Our main contribution is new algorithms for obtaining dual bounds at a node, for checking node fathoming, presolve and duality gap measurement. Our various procedures are implemented and empirically validated on instances from literature and a new set of hard instances. We also perform comparisons against the triangle splitting method of Boland et al. [\emph{INFORMS Journal on Computing}, \textbf{27} (4), 2015], which is a objective space search algorithm as opposed to our variable space search algorithm. On each of the literature instances, our branch-and-bound is able to compute the entire Pareto set in significantly lesser time. Most of the instances of the harder problem set were not solved by either algorithm in a reasonable time limit, but our algorithm performs better on average on the instances that were solved.Comment: 35 pages, 12 figures. Original preprint at Optimization Online, October 201

Efficient Storage of Pareto Points in Biobjective Mixed Integer Programming

In biobjective mixed integer linear programs (BOMILPs), two linear objectives are minimized over a polyhedron while restricting some of the variables to be integer. Since many of the techniques for finding or approximating the Pareto set of a BOMILP use and update a subset of nondominated solutions, it is highly desirable to efficiently store this subset. We present a new data structure, a variant of a binary tree that takes as input points and line segments in $\R^2$ and stores the nondominated subset of this input. When used within an exact solution procedure, such as branch-and-bound (BB), at termination this structure contains the set of Pareto optimal solutions. We compare the efficiency of our structure in storing solutions to that of a dynamic list which updates via pairwise comparison. Then we use our data structure in two biobjective BB techniques available in the literature and solve three classes of instances of BOMILP, one of which is generated by us. The first experiment shows that our data structure handles up to $10^7$ points or segments much more efficiently than a dynamic list. The second experiment shows that our data structure handles points and segments much more efficiently than a list when used in a BB

Advancing parametric optimization: on multiparametric linear complementarity problems with parameters in general locations

The theory presented in this work merges many concepts from mathematical optimization and real algebraic geometry. When unknown or uncertain data in an optimization problem is replaced with parameters, one obtains a multi-parametric optimization problem whose optimal solution comes in the form of a function of the parameters.The theory and methodology presented in this work allows one to solve both Linear Programs and convex Quadratic Programs containing parameters in any location within the problem data as well as multi-objective optimization problems with any number of convex quadratic or linear objectives and linear constraints. Applications of these classes of problems are extremely widespread, ranging from business and economics to chemical and environmental engineering. Prior to this work, no solution procedure existed for these general classes of problems except for the recently proposed algorithms

Table_3_The paleo-ecological application of mollusks in the calculation of saltwater encroachment and resultant changes in depositional patterns driven by the Anthropocene Marine Transgression.docx

Numerous studies address changes in wetland deposition in response to saltwater encroachment driven by the accelerating rate of sea-level rise, by quantifying temporal changes recovered from a vertical sediment sequence. This is the first landscape scale study, based upon 10 core transects representing the heterogeneity of the Southeast Saline Everglades, Florida. By utilizing the known salinity preferences of molluscan assemblages, a Salinity Index was calculated for each core sequence and the recorded salinity changes identified and dated. Radiometric dating utilizing the 210Pb method provides the rate of sediment accumulation and the date of changes identified in the core. The core transects provide the basis for calculation of the rate of saltwater encroachment by comparing the date of saltwater encroachment and the distance between two cores. Thereby, temporal and spatial changes in other sediment parameters in a landscape can also be quantified, such as organic carbon. This paleo-ecological approach to rapidly changing coastal conditions can be utilized to provide scientists and land managers with a record of the past, rate of changing conditions and provide the basis for predicting the future trajectory of their site. Application of this paleo-ecological approach documented increasing rates of saltwater encroachment associated with accelerating rate of sea-level rise: an average rate of 49.1 between 1895 and 1940, 69.2 between 1940 and 1968, 73 between 1968 and 1995 and 131.1 m/yr between 1995 and 2015. Approximately 1.79 km of saltwater encroachment has occurred since 1995, with three partial reversals because of increased freshwater delivery. Associated with saltwater encroachment are changes in sediment organic carbon, decreasing area of marl production and increasing distribution of mangrove. Although the distance of saltwater encroachment is greater in Florida Bay, both changes in sediment organic carbon and mangrove distribution are much less than in Biscayne Bay coastal basins. This heterogeneity is likely the result of differences in tidal ingress efficiency. At the present rate of saltwater encroachment, the freshwater wetlands are predictably lost within a century.</p

Table_4_The paleo-ecological application of mollusks in the calculation of saltwater encroachment and resultant changes in depositional patterns driven by the Anthropocene Marine Transgression.docx

Numerous studies address changes in wetland deposition in response to saltwater encroachment driven by the accelerating rate of sea-level rise, by quantifying temporal changes recovered from a vertical sediment sequence. This is the first landscape scale study, based upon 10 core transects representing the heterogeneity of the Southeast Saline Everglades, Florida. By utilizing the known salinity preferences of molluscan assemblages, a Salinity Index was calculated for each core sequence and the recorded salinity changes identified and dated. Radiometric dating utilizing the 210Pb method provides the rate of sediment accumulation and the date of changes identified in the core. The core transects provide the basis for calculation of the rate of saltwater encroachment by comparing the date of saltwater encroachment and the distance between two cores. Thereby, temporal and spatial changes in other sediment parameters in a landscape can also be quantified, such as organic carbon. This paleo-ecological approach to rapidly changing coastal conditions can be utilized to provide scientists and land managers with a record of the past, rate of changing conditions and provide the basis for predicting the future trajectory of their site. Application of this paleo-ecological approach documented increasing rates of saltwater encroachment associated with accelerating rate of sea-level rise: an average rate of 49.1 between 1895 and 1940, 69.2 between 1940 and 1968, 73 between 1968 and 1995 and 131.1 m/yr between 1995 and 2015. Approximately 1.79 km of saltwater encroachment has occurred since 1995, with three partial reversals because of increased freshwater delivery. Associated with saltwater encroachment are changes in sediment organic carbon, decreasing area of marl production and increasing distribution of mangrove. Although the distance of saltwater encroachment is greater in Florida Bay, both changes in sediment organic carbon and mangrove distribution are much less than in Biscayne Bay coastal basins. This heterogeneity is likely the result of differences in tidal ingress efficiency. At the present rate of saltwater encroachment, the freshwater wetlands are predictably lost within a century.</p

Table_5_The paleo-ecological application of mollusks in the calculation of saltwater encroachment and resultant changes in depositional patterns driven by the Anthropocene Marine Transgression.docx

Numerous studies address changes in wetland deposition in response to saltwater encroachment driven by the accelerating rate of sea-level rise, by quantifying temporal changes recovered from a vertical sediment sequence. This is the first landscape scale study, based upon 10 core transects representing the heterogeneity of the Southeast Saline Everglades, Florida. By utilizing the known salinity preferences of molluscan assemblages, a Salinity Index was calculated for each core sequence and the recorded salinity changes identified and dated. Radiometric dating utilizing the 210Pb method provides the rate of sediment accumulation and the date of changes identified in the core. The core transects provide the basis for calculation of the rate of saltwater encroachment by comparing the date of saltwater encroachment and the distance between two cores. Thereby, temporal and spatial changes in other sediment parameters in a landscape can also be quantified, such as organic carbon. This paleo-ecological approach to rapidly changing coastal conditions can be utilized to provide scientists and land managers with a record of the past, rate of changing conditions and provide the basis for predicting the future trajectory of their site. Application of this paleo-ecological approach documented increasing rates of saltwater encroachment associated with accelerating rate of sea-level rise: an average rate of 49.1 between 1895 and 1940, 69.2 between 1940 and 1968, 73 between 1968 and 1995 and 131.1 m/yr between 1995 and 2015. Approximately 1.79 km of saltwater encroachment has occurred since 1995, with three partial reversals because of increased freshwater delivery. Associated with saltwater encroachment are changes in sediment organic carbon, decreasing area of marl production and increasing distribution of mangrove. Although the distance of saltwater encroachment is greater in Florida Bay, both changes in sediment organic carbon and mangrove distribution are much less than in Biscayne Bay coastal basins. This heterogeneity is likely the result of differences in tidal ingress efficiency. At the present rate of saltwater encroachment, the freshwater wetlands are predictably lost within a century.</p

Table_2_The paleo-ecological application of mollusks in the calculation of saltwater encroachment and resultant changes in depositional patterns driven by the Anthropocene Marine Transgression.docx

Numerous studies address changes in wetland deposition in response to saltwater encroachment driven by the accelerating rate of sea-level rise, by quantifying temporal changes recovered from a vertical sediment sequence. This is the first landscape scale study, based upon 10 core transects representing the heterogeneity of the Southeast Saline Everglades, Florida. By utilizing the known salinity preferences of molluscan assemblages, a Salinity Index was calculated for each core sequence and the recorded salinity changes identified and dated. Radiometric dating utilizing the 210Pb method provides the rate of sediment accumulation and the date of changes identified in the core. The core transects provide the basis for calculation of the rate of saltwater encroachment by comparing the date of saltwater encroachment and the distance between two cores. Thereby, temporal and spatial changes in other sediment parameters in a landscape can also be quantified, such as organic carbon. This paleo-ecological approach to rapidly changing coastal conditions can be utilized to provide scientists and land managers with a record of the past, rate of changing conditions and provide the basis for predicting the future trajectory of their site. Application of this paleo-ecological approach documented increasing rates of saltwater encroachment associated with accelerating rate of sea-level rise: an average rate of 49.1 between 1895 and 1940, 69.2 between 1940 and 1968, 73 between 1968 and 1995 and 131.1 m/yr between 1995 and 2015. Approximately 1.79 km of saltwater encroachment has occurred since 1995, with three partial reversals because of increased freshwater delivery. Associated with saltwater encroachment are changes in sediment organic carbon, decreasing area of marl production and increasing distribution of mangrove. Although the distance of saltwater encroachment is greater in Florida Bay, both changes in sediment organic carbon and mangrove distribution are much less than in Biscayne Bay coastal basins. This heterogeneity is likely the result of differences in tidal ingress efficiency. At the present rate of saltwater encroachment, the freshwater wetlands are predictably lost within a century.</p

Table_1_The paleo-ecological application of mollusks in the calculation of saltwater encroachment and resultant changes in depositional patterns driven by the Anthropocene Marine Transgression.docx

Numerous studies address changes in wetland deposition in response to saltwater encroachment driven by the accelerating rate of sea-level rise, by quantifying temporal changes recovered from a vertical sediment sequence. This is the first landscape scale study, based upon 10 core transects representing the heterogeneity of the Southeast Saline Everglades, Florida. By utilizing the known salinity preferences of molluscan assemblages, a Salinity Index was calculated for each core sequence and the recorded salinity changes identified and dated. Radiometric dating utilizing the 210Pb method provides the rate of sediment accumulation and the date of changes identified in the core. The core transects provide the basis for calculation of the rate of saltwater encroachment by comparing the date of saltwater encroachment and the distance between two cores. Thereby, temporal and spatial changes in other sediment parameters in a landscape can also be quantified, such as organic carbon. This paleo-ecological approach to rapidly changing coastal conditions can be utilized to provide scientists and land managers with a record of the past, rate of changing conditions and provide the basis for predicting the future trajectory of their site. Application of this paleo-ecological approach documented increasing rates of saltwater encroachment associated with accelerating rate of sea-level rise: an average rate of 49.1 between 1895 and 1940, 69.2 between 1940 and 1968, 73 between 1968 and 1995 and 131.1 m/yr between 1995 and 2015. Approximately 1.79 km of saltwater encroachment has occurred since 1995, with three partial reversals because of increased freshwater delivery. Associated with saltwater encroachment are changes in sediment organic carbon, decreasing area of marl production and increasing distribution of mangrove. Although the distance of saltwater encroachment is greater in Florida Bay, both changes in sediment organic carbon and mangrove distribution are much less than in Biscayne Bay coastal basins. This heterogeneity is likely the result of differences in tidal ingress efficiency. At the present rate of saltwater encroachment, the freshwater wetlands are predictably lost within a century.</p