9.1 Linear programming

Linear programming is a powerful optimization technique in computational geometry. It allows us to maximize or minimize linear functions subject to linear constraints, solving complex geometric problems efficiently. This mathematical approach forms the foundation for many algorithms in resource allocation and spatial analysis.

The simplex algorithm and interior point methods are key tools for solving linear programs. These techniques navigate the feasible region, defined by constraints, to find optimal solutions. Applications in computational geometry include convex hull algorithms, Voronoi diagrams, and intersection detection, showcasing the versatility of linear programming.

Fundamentals of linear programming

• Linear programming forms a crucial foundation in computational geometry, enabling the optimization of linear objective functions subject to linear constraints
• Applies mathematical techniques to solve complex geometric problems efficiently, particularly in areas like resource allocation and spatial analysis
• Serves as a powerful tool for modeling and solving various geometric optimization problems in computational geometry

Definition and components

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• Mathematical optimization technique maximizing or minimizing a linear objective function subject to linear equality and inequality constraints
• Decision variables represent quantities to be determined, forming the basis of the optimization problem
• Standard form expresses the problem as maximizing $c^T x$ subject to $Ax \leq b$ and $x \geq 0$, where $x$ is the vector of decision variables
• Utilizes matrix notation for compact representation of large-scale problems

Objective function

• Linear expression defining the goal of the optimization problem (maximize profit, minimize cost)
• Represented mathematically as $z = c_1x_1 + c_2x_2 + ... + c_nx_n$, where $c_i$ are coefficients and $x_i$ are decision variables
• Determines the direction of optimization (maximization or minimization)
• Plays a crucial role in guiding the search for optimal solutions in geometric problems

Constraints and feasible region

• Linear inequalities or equalities that restrict the possible values of decision variables
• Geometric interpretation forms a convex polytope in n-dimensional space
• Feasible region comprises all points satisfying all constraints simultaneously
• Bounded feasible regions ensure the existence of optimal solutions
• Unbounded regions may lead to infeasible or unbounded optimization problems

Geometric interpretation

• Provides visual representation of linear programming problems in two or three dimensions
• Facilitates intuitive understanding of solution spaces and optimization processes
• Connects abstract mathematical concepts to tangible geometric structures

Graphical representation

• Plots constraints as lines or planes in a coordinate system
• Shaded area represents the feasible region where all constraints are satisfied
• Objective function appears as a family of parallel lines or planes
• Optimal solution occurs at a vertex or edge of the feasible region
• Limited to two or three dimensions for visual representation

Feasible solutions vs optimal solutions

• Feasible solutions satisfy all constraints but may not maximize or minimize the objective function
• Optimal solutions achieve the best possible value of the objective function while remaining feasible
• Multiple optimal solutions can exist when the objective function aligns with an edge of the feasible region
• Infeasible problems have no solutions satisfying all constraints simultaneously
• Unbounded problems have infinitely improving solutions without a finite optimum

Convex polyhedra

• Geometric shape formed by the intersection of finitely many half-spaces
• Represents the feasible region in higher-dimensional linear programming problems
• Fundamental property ensures that local optima are also global optima
• Extreme points (vertices) of the polyhedron correspond to basic feasible solutions
• Edges connect adjacent extreme points, representing potential paths for optimization algorithms

Simplex algorithm

• Efficient iterative method for solving linear programming problems
• Moves along the edges of the feasible region to improve the objective function value
• Widely used in practice due to its reliability and effectiveness in most real-world problems

Basic principle

• Starts at an initial basic feasible solution (typically a vertex of the feasible region)
• Iteratively moves to adjacent vertices that improve the objective function value
• Terminates when no further improvement is possible or unboundedness is detected
• Exploits the fact that optimal solutions occur at extreme points of the feasible region
• Guarantees finding an optimal solution in finite steps for non-degenerate problems

Pivot operations

• Algebraic procedure for moving from one basic feasible solution to another
• Involves selecting a non-basic variable to enter the basis and a basic variable to leave
• Utilizes Gaussian elimination to update the tableau representation of the problem
• Determines the direction and magnitude of movement along edges of the feasible region
• Efficiency of the algorithm depends on careful selection of pivot elements

Degeneracy and cycling

• Degeneracy occurs when a basic feasible solution has one or more basic variables equal to zero
• Can lead to stalling or cycling, where the algorithm revisits the same sequence of solutions
• Bland's rule prevents cycling by imposing a specific order for selecting entering and leaving variables
• Lexicographic method provides an alternative approach to handle degeneracy
• Perturbation techniques slightly modify the problem to avoid degenerate situations

Duality in linear programming

• Establishes a relationship between pairs of related linear programming problems
• Provides powerful insights into solution properties and sensitivity analysis
• Plays a crucial role in developing efficient algorithms and understanding problem structure

Primal vs dual problems

• Every linear program (primal) has an associated dual problem
• Primal maximization problem corresponds to a dual minimization problem and vice versa
• Variables in the dual problem correspond to constraints in the primal problem
• Constraints in the dual problem correspond to variables in the primal problem
• Objective function coefficients and right-hand side values swap roles between primal and dual

Weak duality theorem

• States that the objective value of any feasible solution to the primal problem is less than or equal to the objective value of any feasible solution to the dual problem
• Provides bounds on optimal solutions without solving the problems completely
• Useful for developing approximation algorithms and establishing optimality certificates
• Holds regardless of whether optimal solutions exist for either problem
• Forms the basis for more advanced duality results and algorithms

Strong duality theorem

• Asserts that if either the primal or dual problem has an optimal solution, then both have optimal solutions, and their optimal objective values are equal
• Provides a powerful tool for verifying optimality of solutions
• Enables solving one problem indirectly by solving its dual counterpart
• Forms the foundation for complementary slackness conditions
• Crucial in developing interior point methods and sensitivity analysis techniques

Applications in computational geometry

• Linear programming serves as a versatile tool for solving various geometric problems
• Enables efficient algorithms for fundamental computational geometry tasks
• Provides a framework for optimizing geometric structures and relationships

Convex hull algorithms

• Utilizes linear programming to find the smallest convex set containing a given set of points
• Implements incremental algorithms that add points one by one, updating the convex hull
• Applies duality to transform the problem into finding intersections of half-planes
• Achieves optimal time complexity for two-dimensional convex hull computation
• Extends to higher dimensions using more sophisticated linear programming techniques

Voronoi diagrams

• Constructs a partition of space based on the nearest-neighbor rule using linear programming
• Formulates the problem as finding the intersection of half-spaces defined by bisectors
• Employs duality to transform the problem into computing a convex hull in dual space
• Enables efficient computation of nearest neighbor queries and proximity structures
• Finds applications in various fields (computational biology, computer graphics, robotics)

Intersection detection

• Applies linear programming to determine whether geometric objects intersect
• Formulates intersection problems as feasibility questions in linear programming
• Utilizes the simplex algorithm or interior point methods to solve the resulting LP problems
• Enables efficient algorithms for detecting intersections between convex polytopes
• Extends to more complex geometric objects through decomposition into convex pieces

Interior point methods

• Alternative approach to solving linear programming problems
• Moves through the interior of the feasible region rather than along its boundary
• Achieves polynomial time complexity, making it competitive with the simplex algorithm

Barrier functions

• Incorporate constraints into the objective function using penalty terms
• Transform constrained optimization problems into unconstrained problems
• Logarithmic barrier functions commonly used in linear programming
• Create a smooth approximation of the original problem's feasible region
• Enable the use of efficient unconstrained optimization techniques

Central path

• Sequence of points in the interior of the feasible region approaching the optimal solution
• Defined by the set of solutions to a parameterized system of equations
• Provides a trajectory for interior point methods to follow towards optimality
• Exhibits desirable numerical properties and convergence behavior
• Forms the basis for path-following interior point algorithms

Polynomial time complexity

• Interior point methods achieve worst-case polynomial time complexity for linear programming
• Karmarkar's algorithm (1984) sparked renewed interest in interior point methods
• Primal-dual methods offer improved practical performance and theoretical guarantees
• Compete with the simplex algorithm, especially for large-scale problems
• Enable efficient solution of semidefinite and second-order cone programming problems

Sensitivity analysis

• Studies how changes in problem parameters affect the optimal solution
• Provides valuable insights for decision-making and understanding problem structure
• Utilizes duality theory and the properties of basic feasible solutions

Parametric linear programming

• Analyzes how optimal solutions change as a function of one or more parameters
• Traces the path of optimal solutions as parameters vary continuously
• Identifies critical values where the optimal basis changes
• Enables efficient reoptimization for small parameter changes
• Finds applications in robust optimization and multi-objective programming

Shadow prices

• Dual variables associated with constraints in the primal problem
• Represent the rate of change in the objective function value per unit change in the right-hand side of a constraint
• Provide economic interpretation of constraint sensitivity
• Help identify binding constraints and potential areas for improvement
• Used in resource allocation and pricing decisions

Range of optimality

• Determines the range of values for coefficients or right-hand sides that maintain the current optimal solution
• Identifies how much a parameter can change before a new optimal solution is required
• Utilizes the optimal tableau and reduced costs to compute allowable ranges
• Provides insights into the stability and robustness of optimal solutions
• Helps in scenario analysis and decision-making under uncertainty

Integer linear programming

• Extends linear programming to problems where some or all variables must take integer values
• Introduces combinatorial complexity, making the problem NP-hard
• Develops specialized algorithms to find optimal integer solutions efficiently

Branch and bound

• Systematic enumeration method for solving integer linear programming problems
• Divides the problem into subproblems by branching on integer variables
• Uses linear programming relaxations to provide bounds on optimal solutions
• Prunes branches that cannot lead to better solutions than the current best
• Implements various branching and node selection strategies to improve efficiency

Cutting plane methods

• Iteratively adds constraints (cuts) to tighten the linear programming relaxation
• Generates valid inequalities that separate fractional solutions from integer feasible solutions
• Gomory cuts provide a systematic way to generate cutting planes
• Combines with branch and bound to create branch-and-cut algorithms
• Effective for certain classes of integer programming problems (facility location, traveling salesman)

Relaxation techniques

• Replaces hard integer constraints with easier-to-solve continuous relaxations
• Linear programming relaxation removes integrality constraints
• Lagrangian relaxation moves difficult constraints into the objective function
• Semidefinite relaxation lifts the problem to a higher-dimensional space
• Provides bounds and approximate solutions for integer programming problems

Linear programming solvers

• Software packages designed to efficiently solve linear programming problems
• Implement various algorithms and techniques to handle different problem sizes and structures
• Play a crucial role in applying linear programming to real-world computational geometry problems

Commercial vs open-source

• Commercial solvers (CPLEX, Gurobi) offer high performance and extensive support
• Open-source alternatives (GLPK, CLP) provide free access and customization options
• Hybrid approaches combine open-source modeling languages with commercial solvers
• Performance differences can be significant for large-scale or challenging problems
• Choice depends on problem size, complexity, and available resources

Numerical stability issues

• Addresses challenges arising from finite-precision arithmetic in computers
• Implements scaling techniques to improve the conditioning of problem matrices
• Utilizes iterative refinement to enhance the accuracy of computed solutions
• Employs basis update procedures to maintain numerical stability in the simplex method
• Implements safeguards against cycling and stalling in degenerate problems

Large-scale optimization

• Develops specialized techniques for problems with millions of variables or constraints
• Exploits problem structure (sparsity, network flow) to reduce memory and computation requirements
• Implements parallel and distributed algorithms to leverage multi-core processors and clusters
• Utilizes column generation and decomposition methods for problems with special structure
• Applies presolve techniques to reduce problem size and improve solver performance

Advanced topics

• Explores extensions and generalizations of linear programming
• Addresses more complex optimization problems arising in computational geometry and related fields
• Develops specialized algorithms and theoretical frameworks for advanced optimization tasks

Semidefinite programming

• Optimizes linear functions over the cone of positive semidefinite matrices
• Generalizes linear programming to matrix variables
• Finds applications in relaxations of combinatorial optimization problems
• Enables efficient approximation algorithms for various NP-hard problems
• Utilizes interior point methods adapted for semidefinite constraints

Quadratic programming

• Optimizes quadratic objective functions subject to linear constraints
• Includes important special cases (convex quadratic programming, quadratically constrained quadratic programming)
• Applies to various computational geometry problems (least squares fitting, support vector machines)
• Develops specialized algorithms (active set methods, interior point methods) for efficient solution
• Extends to more general nonlinear programming problems

Multi-objective linear programming

• Simultaneously optimizes multiple, possibly conflicting, linear objective functions
• Introduces the concept of Pareto optimality for trade-offs between objectives
• Develops methods for generating and exploring the set of non-dominated solutions
• Applies to decision-making problems in computational geometry and beyond
• Utilizes techniques (weighted sum method, ε-constraint method) to transform multi-objective problems into single-objective problems