76 research outputs found
On linear, fractional, and submodular optimization
In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree
Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications
Submodularity is an important concept in integer and combinatorial
optimization. A classical submodular set function models the utility of
selecting homogenous items from a single ground set, and such selections can be
represented by binary variables. In practice, many problem contexts involve
choosing heterogenous items from more than one ground set or selecting multiple
copies of homogenous items, which call for extensions of submodularity. We
refer to the optimization problems associated with such generalized notions of
submodularity as Generalized Submodular Optimization (GSO). GSO is found in
wide-ranging applications, including infrastructure design, healthcare, online
marketing, and machine learning. Due to the often highly nonlinear (even
non-convex and non-concave) objective function and the mixed-integer decision
space, GSO is a broad subclass of challenging mixed-integer nonlinear
programming problems. In this tutorial, we first provide an overview of
classical submodularity. Then we introduce two subclasses of GSO, for which we
present polyhedral theory for the mixed-integer set structures that arise from
these problem classes. Our theoretical results lead to efficient and versatile
exact solution methods that demonstrate their effectiveness in practical
problems using real-world datasets
Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization
Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes
Capacity Planning in Stable Matching
We introduce the problem of jointly increasing school capacities and finding
a student-optimal assignment in the expanded market. Due to the impossibility
of efficiently solving the problem with classical methods, we generalize
existent mathematical programming formulations of stability constraints to our
setting, most of which result in integer quadratically-constrained programs. In
addition, we propose a novel mixed-integer linear programming formulation that
is exponentially large on the problem size. We show that its stability
constraints can be separated by exploiting the objective function, leading to
an effective cutting-plane algorithm. We conclude the theoretical analysis of
the problem by discussing some mechanism properties. On the computational side,
we evaluate the performance of our approaches in a detailed study, and we find
that our cutting-plane method outperforms our generalization of existing
mixed-integer approaches. We also propose two heuristics that are effective for
large instances of the problem. Finally, we use the Chilean school choice
system data to demonstrate the impact of capacity planning under stability
conditions. Our results show that each additional seat can benefit multiple
students and that we can effectively target the assignment of previously
unassigned students or improve the assignment of several students through
improvement chains. These insights empower the decision-maker in tuning the
matching algorithm to provide a fair application-oriented solution
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Approximation Algorithms for Independence Systems
In this thesis, we study three maximization problems over independence systems. • Chapter 2 – Weighted k-Set Packing is a fundamental combinatorial optimization problem that captures matching problems in graphs and hypergraphs. For over 20 years Berman’s algorithm stood as the state-of-the-art approximation algorithm for this problem, until Neuwohner’s recent improvements. Our focus is on the value k = 3 which is well motivated from theory and practice, and for which improvements are arguably the hardest. We largely improve upon her approximation, by giving an algorithm that yields state-of-the-art results. Our techniques are simple and naturally expand upon Berman’s analysis. Our analysis holds for any value of k with greater improvements over Berman’s result as k grows. • Chapter 3 – We continue the study of the weighted k-set packing problem. Building on Chapter 2, we reach the tightest approximation factor possible for k = 3, and k ≥ 7 using our techniques. As a consequence, we improve over all the results in Chapter 2. In particular, we obtain √3, and k/2 -approximation for k = 3 and k ≥ 7 respectively. Our result for k ≥ 7 is in fact analogous to that of Hurkens and Schrijver who obtained the same approximation factor for the unweighted problem. • Chapter 4 – We present improved multipass streaming algorithms for maximizing monotone and arbitrary submodular functions over independence systems. Our result demonstrates that the simple local-search algorithm for maximizing a monotone sub- modular function can be efficiently simulated using a few passes over the dataset. Our results improve the number of passes needed compared to the state-of-the-art. • Chapter 5 – We conclude the thesis by presenting improved approximation algorithms for Sparse Least-Square Estimation, Bayesian A-optimal Design, and Column Subset Selection over a matroid constraint. At the heart of this chapter is the demonstration of a new property that considered applications satisfy. We call it: β-weak submodularity. We leverage this property to derive new algorithms with strengthened guarantees. The notion of β-weak submodularity is of independent interest and we believe that it will have further use in machine learning and statistics
On Constrained Mixed-Integer DR-Submodular Minimization
DR-submodular functions encompass a broad class of functions which are
generally non-convex and non-concave. We study the problem of minimizing any
DR-submodular function, with continuous and general integer variables, under
box constraints and possibly additional monotonicity constraints. We propose
valid linear inequalities for the epigraph of any DR-submodular function under
the constraints. We further provide the complete convex hull of such an
epigraph, which, surprisingly, turns out to be polyhedral. We propose a
polynomial-time exact separation algorithm for our proposed valid inequalities,
with which we first establish the polynomial-time solvability of this class of
mixed-integer nonlinear optimization problems
Social Media Influencers- A Review of Operations Management Literature
This literature review provides a comprehensive survey of research on Social Media
Influencers (SMIs) across the fields of SMIs in marketing, seeding strategies, influence
maximization and applications of SMIs in society. Specifically, we focus on examining the
methods employed by researchers to reach their conclusions. Through our analysis, we
identify opportunities for future research that align with emerging areas and unexplored
territories related to theory, context, and methodology. This approach offers a fresh
perspective on existing research, paving the way for more effective and impactful studies in
the future. Additionally, gaining a deeper understanding of the underlying principles and
methodologies of these concepts enables more informed decision-making when
implementing these strategie
Efficient streaming algorithms for maximizing monotone DR-submodular function on the integer lattice
In recent years, the issue of maximizing submodular functions has attracted much interest from research communities. However, most submodular functions are specified in a set function. Meanwhile, recent advancements have been studied for maximizing a diminishing return submodular (DR-submodular) function on the integer lattice. Because plenty of publications show that the DR-submodular function has wide applications in optimization problems such as sensor placement impose problems, optimal budget allocation, social network, and especially machine learning. In this research, we propose two main streaming algorithms for the problem of maximizing a monotone DR-submodular function under cardinality constraints. Our two algorithms, which are called StrDRS1 and StrDRS2, have (1/2 - epsilon) , (1 - 1 /e - epsilon) of approximation ratios and O(n/epsilon log(log B/epsilon ) log k), O(n/epsilon log B), respectively. We conducted several experiments to investigate the performance of our algorithms based on the budget allocation problem over the bipartite influence model, an instance of the monotone submodular function maximization problem over the integer lattice. The experimental results indicate that our proposed algorithms not only provide solutions with a high value of the objective function, but also outperform the state-of-the-art algorithms in terms of both the number of queries and the running time.Web of Science1020art. no. 377
- …