+publisher:"Georgia Tech" +contributor:("Dey, Santanu S.")

1. Zink, Daniel. A reduction framework for approximate extended formulations and a faster algorithm for convex optimization.

Degree: PhD, Industrial and Systems Engineering, 2017, Georgia Tech

URL: http://hdl.handle.net/1853/58274

► Linear programming (LP) and semidefinite programming (SDP) are among the most important tools in Operations Research and Computer Science. In this work we study the… (more)

▼ Linear programming (LP) and semidefinite programming (SDP) are among the most important tools in Operations Research and Computer Science. In this work we study the limitations of LPs and SDPs by providing lower bounds on the size of (approximate) linear and semidefinite programming formulations of combinatorial optimization problems. The hardness of (approximate) linear optimization implied by these lower bounds motivates the lazification technique for conditional gradient type algorithms. This technique allows us to replace (approximate) linear optimization in favor of a much weaker subroutine, achieving significant performance improvement in practice. We can summarize the main contributions as follows: (i) Reduction framework for LPs and SDPs: We present a new view on extended formulations that does not require an initial encoding of a combinatorial problem as a linear or semidefinite program. This new view allows us to define a purely combinatorial reduction framework transferring lower bounds on the size of exact and approximate LP and SDP formulations between different problems. Using our framework we show new LP and SDP lower bounds for a large variety of problems including Vertex Cover, various (binary and non-binary) constraint satisfaction problems as well as multiple optimization versions of Graph-Isomorphism. (ii) Lazification technique for Conditional Gradient algorithms: In Convex Programming conditional gradient type algorithms (also known as Frank-Wolfe type methods) are very important in theory as well as in practice due to their simplicity and fast convergence. We show how we can eliminate the linear optimization step performed in every iteration of Frank-Wolfe type methods and instead use a weak separation oracle. This oracle is significantly faster in practice and enables caching for additional improvements in speed and the sparsity of the obtained solutions. Advisors/Committee Members: Pokutta, Sebastian (advisor), Blekherman, Grigoriy (committee member), Dey, Santanu S. (committee member), Lan, Guanghui (committee member), Vempala, Santosh (committee member).

Subjects/Keywords: Extended formulations; Linear programming; Semidefinite programming; Approximations; Convex optimization; Frank-Wolfe method; Conditional gradients

1 more image…

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Zink, D. (2017). A reduction framework for approximate extended formulations and a faster algorithm for convex optimization. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/58274

Chicago Manual of Style (16^th Edition):

Zink, Daniel. “A reduction framework for approximate extended formulations and a faster algorithm for convex optimization.” 2017. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/58274.

MLA Handbook (7^th Edition):

Zink, Daniel. “A reduction framework for approximate extended formulations and a faster algorithm for convex optimization.” 2017. Web. 12 Dec 2019.

Vancouver:

Zink D. A reduction framework for approximate extended formulations and a faster algorithm for convex optimization. [Internet] [Doctoral dissertation]. Georgia Tech; 2017. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/58274.

Council of Science Editors:

Zink D. A reduction framework for approximate extended formulations and a faster algorithm for convex optimization. [Doctoral Dissertation]. Georgia Tech; 2017. Available from: http://hdl.handle.net/1853/58274

Georgia Tech

2. Kocuk, Burak. Global optimization methods for optimal power flow and transmission switching problems in electric power systems.

Degree: PhD, Industrial and Systems Engineering, 2016, Georgia Tech

URL: http://hdl.handle.net/1853/55633

► Power engineering is concerned with the generation, transmission, and distribution of electricity over electric power network, which is arguably one of the largest engineering systems… (more)

Subjects/Keywords: Global optimization; Conic programming; Mixed-integer nonlinear programming; Power systems; Optimal power flow; Transmission switching

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Kocuk, B. (2016). Global optimization methods for optimal power flow and transmission switching problems in electric power systems. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/55633

Chicago Manual of Style (16^th Edition):

Kocuk, Burak. “Global optimization methods for optimal power flow and transmission switching problems in electric power systems.” 2016. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/55633.

MLA Handbook (7^th Edition):

Kocuk, Burak. “Global optimization methods for optimal power flow and transmission switching problems in electric power systems.” 2016. Web. 12 Dec 2019.

Vancouver:

Kocuk B. Global optimization methods for optimal power flow and transmission switching problems in electric power systems. [Internet] [Doctoral dissertation]. Georgia Tech; 2016. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/55633.

Council of Science Editors:

Kocuk B. Global optimization methods for optimal power flow and transmission switching problems in electric power systems. [Doctoral Dissertation]. Georgia Tech; 2016. Available from: http://hdl.handle.net/1853/55633

3. Xie, Weijun. Relaxations and approximations of chance constrained stochastic programs.

Degree: PhD, Industrial and Systems Engineering, 2017, Georgia Tech

URL: http://hdl.handle.net/1853/58678

► A chance constrained stochastic programming (CCSP) problem involves constraints with random parameters that are required to be satisfied with a prespecified probability threshold. Such constraints… (more)

▼ A chance constrained stochastic programming (CCSP) problem involves constraints with random parameters that are required to be satisfied with a prespecified probability threshold. Such constraints are used to model reliability requirements in a variety of application areas such as finance, energy, service and manufacturing. Except under very special conditions, chance constraints impart severe nonconvexities making the optimization problem extremely difficult. Moreover, in many cases, the probability distribution of the random parameters is not fully specified giving rise to additional difficulties. This thesis makes several contributions towards alleviating these two difficulties in CCSP. In the first part of this thesis we consider CCSP problems with finitely supported probability distributions. Such problems can be reformulated as mixed integer programming (MIP) problems. We propose two new efficiently solvable Lagrangian dual problems for these problems, and show that their corresponding primal formulations lead to MIP formulations that can be stronger than traditional formulations. We next study a well-known family of cuts for these problems known as quantile cuts. We show that the closure of the infinite family of all quantile cuts has a finite description, and a recursive application of quantile closure operations recovers the convex hull of the nonconvex chance constrained set in the limit. Furthermore, we show that in the pure integer setting, the convergence is finite. Our final result in this part concerns with approximation algorithms for CCSP. We first prove that CCSP is constant factor inapproximable in general. On the other hand, for CCSP problems involving covering type constraints, we prove a bicriteria approximation result where, by relaxing the required probability threshold by a constant factor, we can provide a constant factor approximation algorithm. In the second part of the thesis we consider distributionally robust chance constrained problems (DRCCPs) where the chance constraint is required to hold for all probability distributions of the random constraint parameters from a given ambiguity set. First, we study DRCCPs involving convex nonlinear uncertain constraints and ambiguity sets specified by convex moment constraints. We develop deterministic reformulations of such DRCCPs and identify conditions under which such reformulations are convex. Our results generalize and extend several existing results on convex reformulations of DRCCPs. Next, we apply the proposed reformulation scheme to an optimal power flow problem involving uncertainty stemming from renewable power generation. In particular, we develop a convex programming approach for a distributionally robust chance constrained optimal power flow model that ensures low probability of violating upper and lower limits of a line/bus capacity under a wide family of distributions of uncertain renewable generation. Finally, we study a conservative approximation - referred to as a Bonferroni approximation - of a joint chance constraint,… Advisors/Committee Members: Ahmed, Shabbir (advisor), Dey, Santanu S (committee member), Luedtke, James R (committee member), Shapiro, Alexander (committee member), Nemirovski, Arkadi (committee member).

Subjects/Keywords: chance constraint; approximation algorithm; Lagrangian relaxation; distributionally robust; convex program

1 more image…

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Xie, W. (2017). Relaxations and approximations of chance constrained stochastic programs. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/58678

Chicago Manual of Style (16^th Edition):

Xie, Weijun. “Relaxations and approximations of chance constrained stochastic programs.” 2017. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/58678.

MLA Handbook (7^th Edition):

Xie, Weijun. “Relaxations and approximations of chance constrained stochastic programs.” 2017. Web. 12 Dec 2019.

Vancouver:

Xie W. Relaxations and approximations of chance constrained stochastic programs. [Internet] [Doctoral dissertation]. Georgia Tech; 2017. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/58678.

Council of Science Editors:

Xie W. Relaxations and approximations of chance constrained stochastic programs. [Doctoral Dissertation]. Georgia Tech; 2017. Available from: http://hdl.handle.net/1853/58678

4. Yu, Jiajin. Optimization and separation for structured submodular functions with constraints.

Degree: PhD, Computer Science, 2015, Georgia Tech

URL: http://hdl.handle.net/1853/53517

► Various kinds of optimization problems involve nonlinear functions of binary variables that exhibit a property of diminishing marginal returns. Such a property is known as… (more)

▼ Various kinds of optimization problems involve nonlinear functions of binary variables that exhibit a property of diminishing marginal returns. Such a property is known as submodularity. Vast amount of work has been devoted to the problem of submodular optimization. In this thesis, we exploit structural information for several classes of submodular optimization problems. We strive for polynomial time algorithms with improved approximation ratio and strong mixed-integer linear formulations of mixed-integer non-linear programs where the epigraph and hypograph of submodular functions of a specific form appear as a substructure together with other side constraints. In Chapter 2, we develop approximation algorithms for the expected utility knapsack problem. We use the sample average approximation framework to approximate the stochastic problem as a deterministic knapsack-constrained submodular maximization problem, and then use an approximation algorithm to solve the deterministic counterpart. We show that a polynomial number of samples are enough for a deterministic approximation that is close in relative error. Then, exploiting the strict monotonicity of typical utility functions, we present an algorithm that maximizes an increasing submodular function over a knapsack constraint with approximation ratio better than the classical (1-1/e) ratio. In Chapter 3, we present polyhedral results for the expected utility knapsack problem. We study a mixed-integer nonlinear set that is the hypograph of f(a'x) together together with a knapsack constraint. We propose a family of inequalities for the convex hull of the nonlinear set by exploiting both the structure of the submodular function f(a'x) and the knapsack constraint. Effectiveness of the proposed inequalities is shown by computational experiments on expected utility maximization problem with budget constraint using a branch-and-cut framework. In Chapter 4, we study a mixed-integer nonlinear set that is the epigraph of f(a'x) together with a cardinality constraint. This mixed-integer nonlinear set arises as a substructure in various constrained submodular minimization problems. We develop a strong linear formulation of the convex hull of the nonlinear set by exploiting both the submodularity of f(a'x) and the cardinality constraint. We provide a full description of the convex hull of the nonlinear set when the vector a has identical components. We also develop a family of facet-defining inequalities when the vector a has nonidentical components. We demonstrate the effectiveness of the proposed inequalities by solving mean-risk knapsack problems using a branch-and-cut framework. Advisors/Committee Members: Ahmed, Shabbir (advisor), Dey, Santanu S. (committee member), Liption, Richard J. (committee member), Pokutta, Sebastian (committee member), Tetali, Prasad (committee member).

Subjects/Keywords: Submodular optimization; Mixed-integer optimization

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Yu, J. (2015). Optimization and separation for structured submodular functions with constraints. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/53517

Chicago Manual of Style (16^th Edition):

Yu, Jiajin. “Optimization and separation for structured submodular functions with constraints.” 2015. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/53517.

MLA Handbook (7^th Edition):

Yu, Jiajin. “Optimization and separation for structured submodular functions with constraints.” 2015. Web. 12 Dec 2019.

Vancouver:

Yu J. Optimization and separation for structured submodular functions with constraints. [Internet] [Doctoral dissertation]. Georgia Tech; 2015. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/53517.

Council of Science Editors:

Yu J. Optimization and separation for structured submodular functions with constraints. [Doctoral Dissertation]. Georgia Tech; 2015. Available from: http://hdl.handle.net/1853/53517

5. Moran Ramirez, Diego Alejandro. Fundamental properties of convex mixed-integer programs.

Degree: PhD, Industrial and Systems Engineering, 2014, Georgia Tech

URL: http://hdl.handle.net/1853/52309

► In this Ph.D. dissertation research, we lay the mathematical foundations of various fundamental concepts in convex mixed-integer programs (MIPs), that is, optimization problems where all… (more)

▼ In this Ph.D. dissertation research, we lay the mathematical foundations of various fundamental concepts in convex mixed-integer programs (MIPs), that is, optimization problems where all the decision variables belong to a given convex set and, in addition, a subset of them are required to be integer. In particular, we study properties of their feasible region and properties of cutting planes. The main contribution of this work is the extension of several fundamental results from the theory of linear MIPs to the case of convex MIPs. In the first part, we study properties of general closed convex sets that determine the closedness and polyhedrality of their integer hulls. We first present necessary and sufficient conditions for the integer hull of a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones, strictly convex sets, and sets containing integer points in their interior. We then present a sufficient condition for the integer hulls of general convex sets to be polyhedra. This result generalizes the well-known result due to Meyer in the case of linear MIPs. Under a simple technical assumption, we show that these sufficient conditions are also necessary for the integer hull of general convex sets to be polyhedra. In the second part, we apply the previous results to mixed-integer second order conic programs (MISOCPs), a special case of nonlinear convex MIPs. We show that there exists a polynomial time algorithm to check the closedness of the mixed integer hulls of simple MISOCPs. Moreover, in the special case of pure integer problems, we present sufficient conditions for verifying the closedness of the integer hull of intersection of simple MISOCPs that can also be checked in polynomial time. In the third part, we present an extension of the duality theory for linear MIPs to the case of conic MIPs. In particular, we construct a subadditive dual to conic MIPs. Under a simple condition on the primal problem, we are able to prove strong duality. In the fourth part, we study properties of maximal S-free convex sets, where S is a subset of integers contained in an arbitrary convex set. An S-free convex set is a convex set not containing any points of S in its interior. In this part, we show that maximal S-free convex sets are polyhedra and discuss some properties of these sets. In the fifth part, we study some generalizations of the split closure in the case of linear MIPs. Split cuts form a well-known class of valid inequalities for linear MIPs. Cook et al. (1990) showed that the split closure of a rational polyhedron - that is, the set of points in the polyhedron satisfying all split cuts - is again a polyhedron. In this thesis, we extend this result from a single rational polyhedron to the union of a finite number of rational polyhedra. We also show how this result can be used to prove that some generalizations of split cuts, namely cross cuts, also yield closures that are rational polyhedra. Advisors/Committee Members: Dey, Santanu S. (advisor), Ahmed, Shabbir (committee member), Nemirovski, Arkadi (committee member), Cook, William (committee member), Gunluk, Oktay (committee member), Vielma, Juan Pablo (committee member).

Subjects/Keywords: Integer programming; Cutting planes; Convex hull; Integer hull; Optimization; Split cuts

10 more images…

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Moran Ramirez, D. A. (2014). Fundamental properties of convex mixed-integer programs. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/52309

Chicago Manual of Style (16^th Edition):

Moran Ramirez, Diego Alejandro. “Fundamental properties of convex mixed-integer programs.” 2014. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/52309.

MLA Handbook (7^th Edition):

Moran Ramirez, Diego Alejandro. “Fundamental properties of convex mixed-integer programs.” 2014. Web. 12 Dec 2019.

Vancouver:

Moran Ramirez DA. Fundamental properties of convex mixed-integer programs. [Internet] [Doctoral dissertation]. Georgia Tech; 2014. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/52309.

Council of Science Editors:

Moran Ramirez DA. Fundamental properties of convex mixed-integer programs. [Doctoral Dissertation]. Georgia Tech; 2014. Available from: http://hdl.handle.net/1853/52309

6. He, Qie. Topics in discrete optimization: models, complexity and algorithms.

Degree: PhD, Industrial and Systems Engineering, 2013, Georgia Tech

URL: http://hdl.handle.net/1853/50237

► In this dissertation we examine several discrete optimization problems through the perspectives of modeling, complexity and algorithms. We first provide a probabilistic comparison of split… (more)

▼ In this dissertation we examine several discrete optimization problems through the perspectives of modeling, complexity and algorithms. We first provide a probabilistic comparison of split and type 1 triangle cuts for mixed-integer programs with two rows and two integer variables in terms of cut coefficients and volume cutoff. Under a specific probabilistic model of the problem parameters, we show that for the above measure, the probability that a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut is better than a split cut. The analysis also suggests some guidelines on when type 1 triangle cuts are likely to be more effective than split cuts and vice versa. We next study a minimum concave cost network flow problem over a grid network. We give a polytime algorithm to solve this problem when the number of echelons is fixed. We show that the problem is NP-hard when the number of echelons is an input parameter. We also extend our result to grid networks with backward and upward arcs. Our result unifies the complexity results for several models in production planning and green recycling including the lot-sizing model, and gives the first polytime algorithm for some problems whose complexities were not known before. Finally, we examine how much complexity randomness will bring to a simple combinatorial optimization problem. We study a problem called the sell or hold problem (SHP). SHP is to sell k out of n indivisible assets over two stages, with known first-stage prices and random second-stage prices, to maximize the total expected revenue. Although the deterministic version of SHP is trivial to solve, we show that SHP is NP-hard when the second-stage prices are realized as a finite set of scenarios. We show that SHP is polynomially solvable when the number of scenarios in the second stage is constant. A max{1/2,k/n}-approximation algorithm is presented for the scenario-based SHP. Advisors/Committee Members: Ahmed, Shabbir (advisor), Nemhauser, George L. (advisor), Cook, William J. (committee member), Dey, Santanu S. (committee member), Thomas, Robin (committee member).

Subjects/Keywords: Integer programming; Combinatorial optimization; Stochastic programming; Network flow; Production planning; Computational complexity; Mathematical optimization; Integer programming

2 more images…

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

He, Q. (2013). Topics in discrete optimization: models, complexity and algorithms. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/50237

Chicago Manual of Style (16^th Edition):

He, Qie. “Topics in discrete optimization: models, complexity and algorithms.” 2013. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/50237.

MLA Handbook (7^th Edition):

He, Qie. “Topics in discrete optimization: models, complexity and algorithms.” 2013. Web. 12 Dec 2019.

Vancouver:

He Q. Topics in discrete optimization: models, complexity and algorithms. [Internet] [Doctoral dissertation]. Georgia Tech; 2013. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/50237.

Council of Science Editors:

He Q. Topics in discrete optimization: models, complexity and algorithms. [Doctoral Dissertation]. Georgia Tech; 2013. Available from: http://hdl.handle.net/1853/50237

7. Qiu, Feng. Probabilistic covering problems.

Degree: PhD, Industrial and Systems Engineering, 2013, Georgia Tech

URL: http://hdl.handle.net/1853/47567

► This dissertation studies optimization problems that involve probabilistic covering constraints. A probabilistic constraint evaluates and requires that the probability that a set of constraints involving… (more)

▼ This dissertation studies optimization problems that involve probabilistic covering constraints. A probabilistic constraint evaluates and requires that the probability that a set of constraints involving random coefficients with known distributions hold satisfy a minimum requirement. A covering constraint involves a linear inequality on non-negative variables with a greater or equal to sign and non-negative coefficients. A variety of applications, such as set cover problems, node/edge cover problems, crew scheduling, production planning, facility location, and machine learning, in uncertain settings involve probabilistic covering constraints. In the first part of this dissertation we consider probabilistic covering linear programs. Using the sampling average approximation (SAA) framework, a probabilistic covering linear program can be approximated by a covering k-violation linear program (CKVLP), a deterministic covering linear program in which at most k constraints are allowed to be violated. We show that CKVLP is strongly NP-hard. Then, to improve the performance of standard mixed-integer programming (MIP) based schemes for CKVLP, we (i) introduce and analyze a coefficient strengthening scheme, (ii) adapt and analyze an existing cutting plane technique, and (iii) present a branching technique. Through computational experiments, we empirically verify that these techniques are significantly effective in improving solution times over the CPLEX MIP solver. In particular, we observe that the proposed schemes can cut down solution times from as much as six days to under four hours in some instances. We also developed valid inequalities arising from two subsets of the constraints in the original formulation. When incorporating them with a modified coefficient strengthening procedure, we are able to solve a difficult probabilistic portfolio optimization instance listed in MIPLIB 2010, which cannot be solved by existing approaches. In the second part of this dissertation we study a class of probabilistic 0-1 covering problems, namely probabilistic k-cover problems. A probabilistic k-cover problem is a stochastic version of a set k-cover problem, which is to seek a collection of subsets with a minimal cost whose union covers each element in the set at least k times. In a stochastic setting, the coefficients of the covering constraints are modeled as Bernoulli random variables, and the probabilistic constraint imposes a minimal requirement on the probability of k-coverage. To account for absence of full distributional information, we define a general ambiguous k-cover set, which is ``distributionally-robust." Using a classical linear program (called the Boolean LP) to compute the probability of events, we develop an exact deterministic reformulation to this ambiguous k-cover problem. However, since the boolean model consists of exponential number of auxiliary variables, and hence not useful in practice, we use two linear program based bounds on the probability that at least k events occur, which can be obtained by… Advisors/Committee Members: Ahmed, Shabbir (Committee Chair), Dey, Santanu S. (Committee Co-Chair), Goldberg, David A. (Committee Member), Johnson, Ellis (Committee Member), Luedtke, James (Committee Member), Nemhauser, George (Committee Member).

Subjects/Keywords: Optimization; Stochastic programming; Chance-constrained program; Mixed-integer program; Probabilistic program; Covering problem; Mathematical optimization; Linear programming

10 more images…

Record Details Similar Records Cite « Share

❌

APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^th Edition):

Qiu, F. (2013). Probabilistic covering problems. (Doctoral Dissertation). Georgia Tech. Retrieved from http://hdl.handle.net/1853/47567

Chicago Manual of Style (16^th Edition):

Qiu, Feng. “Probabilistic covering problems.” 2013. Doctoral Dissertation, Georgia Tech. Accessed December 12, 2019. http://hdl.handle.net/1853/47567.

MLA Handbook (7^th Edition):

Qiu, Feng. “Probabilistic covering problems.” 2013. Web. 12 Dec 2019.

Vancouver:

Qiu F. Probabilistic covering problems. [Internet] [Doctoral dissertation]. Georgia Tech; 2013. [cited 2019 Dec 12]. Available from: http://hdl.handle.net/1853/47567.

Council of Science Editors:

Qiu F. Probabilistic covering problems. [Doctoral Dissertation]. Georgia Tech; 2013. Available from: http://hdl.handle.net/1853/47567

		in
And / Or
		in
And / Or
		in
And / Or
		in

Open Access Theses and Dissertations