SAS/OR(R) 9.2 User's Guide: Mathematical Programming
Acknowledgments- What's New in SAS/OR 9.2
- Using This Book
- Purpose
- Organization
- Typographical Conventions
- Conventions for Examples
- Accessing the SAS/OR Sample Library
- Online Help System and Updates
- Additional Documentation for SAS/OR Software
- Introduction to Optimization
- Overview
- Linear Programming Problems
- Network Problems
- Mixed Integer Linear Problems
- Quadratic Programming Problems
- Nonlinear Problems
- Model Building
- Matrix Generation
- Exploiting Model Structure
- Report Writing
- References
- The INTPOINT Procedure
- Overview: INTPOINT Procedure
- Mathematical Description of NPSC
- Mathematical Description of LP
- The Interior Point Algorithm
- Network Models
- Introduction
- Syntax: INTPOINT Procedure
- Functional Summary
- Dictionary of Options
- PROC INTPOINT Statement
- CAPACITY Statement
- COEF Statement
- COLUMN Statement
- COST Statement
- DEMAND Statement
- HEADNODE Statement
- ID Statement
- LO Statement
- NAME Statement
- NODE Statement
- QUIT Statement
- RHS Statement
- ROW Statement
- RUN Statement
- SUPDEM Statement
- SUPPLY Statement
- TAILNODE Statement
- TYPE Statement
- VAR Statement
- Details: INTPOINT Procedure
- Input Data Sets
- Output Data Sets
- Converting Any PROC INTPOINT Format to an MPS-Format SAS Data Set
- Case Sensitivity
- Loop Arcs
- Multiple Arcs
- Flow and Value Bounds
- Tightening Bounds and Side Constraints
- Reasons for Infeasibility
- Missing S Supply and Missing D Demand Values
- Balancing Total Supply and Total Demand
- How to Make the Data Read of PROC INTPOINT More Efficient
- Stopping Criteria
- Macro Variable _ORINTPO
- Memory Limit
- Examples: INTPOINT Procedure
- Example 2.1: Production, Inventory, Distribution Problem
- Example 2.2: Altering Arc Data
- Example 2.3: Adding Side Constraints
- Example 2.4: Using Constraints and More Alteration to Arc Data
- Example 2.5: Nonarc Variables in the Side Constraints
- Example 2.6: Solving an LP Problem with Data in MPS Format
- Example 2.7: Converting to an MPS-Format SAS Data Set
- References
- The LP Procedure
- Overview: LP Procedure
- Getting Started: LP Procedure
- An Introductory Example
- An Integer Programming Example
- An MPS Format to Sparse Format Conversion Example
- Syntax: LP Procedure
- Functional Summary
- Dictionary of Options
- PROC LP Statement
- COEF Statement
- COL Statement
- ID Statement
- IPIVOT Statement
- PIVOT Statement
- PRINT Statement
- QUIT Statement
- RANGE Statement
- RESET Statement
- RHS Statement
- RHSSEN Statement
- ROW Statement
- RUN Statement
- SHOW Statement
- TYPE Statement
- VAR Statement
- Details: LP Procedure
- Missing Values
- Dense Data Input Format
- Sparse Data Input Format
- Converting Any PROC LP Format to an MPS-Format SAS Data Set
- Converting Standard MPS Format to Sparse Format
- The Reduced Costs, Dual Activities, and Current Tableau
- Macro Variable _ORLP_
- Pricing
- Scaling
- Preprocessing
- Integer Programming
- Sensitivity Analysis
- Range Analysis
- Parametric Programming
- Interactive Facilities
- Memory Management
- Output Data Sets
- Input Data Sets
- Displayed Output
- ODS Table and Variable Names
- Memory Limit
- Examples: LP Procedure
- Example 3.1: An Oil Blending Problem
- Example 3.2: A Sparse View of the Oil Blending Problem
- Example 3.3: Sensitivity Analysis: Changes in Objective Coefficients
- Example 3.4: Additional Sensitivity Analysis
- Example 3.5: Price Parametric Programming for the Oil Blending Problem
- Example 3.6: Special Ordered Sets and the Oil Blending Problem
- Example 3.7: Goal-Programming a Product Mix Problem
- Example 3.8: A Simple Integer Program
- Example 3.9: An Infeasible Problem
- Example 3.10: Restarting an Integer Program
- Example 3.11: Alternative Search of the Branch-and-Bound Tree
- Example 3.12: An Assignment Problem
- Example 3.13: A Scheduling Problem
- Example 3.14: A Multicommodity Transshipment Problem with Fixed Charges
- Example 3.15: Converting to an MPS-Format SAS Data Set
- References
- The NLP Procedure
- Overview: NLP Procedure
- Getting Started: NLP Procedure
- Syntax: NLP Procedure
- Functional Summary
- Dictionary of Options
- PROC NLP Statement
- ARRAY Statement
- BOUNDS Statement
- BY Statement
- CRPJAC Statement
- DECVAR Statement
- GRADIENT Statement
- HESSIAN Statement
- INCLUDE Statement
- JACNLC Statement
- JACOBIAN Statement
- LABEL Statement
- LINCON Statement
- MATRIX Statement
- MIN, MAX, and LSQ Statements
- MINQUAD and MAXQUAD Statements
- NLINCON Statement
- PROFILE Statement
- Program Statements
- Details: NLP Procedure
- Criteria for Optimality
- Optimization Algorithms
- Finite-Difference Approximations of Derivatives
- Hessian and CRP Jacobian Scaling
- Testing the Gradient Specification
- Termination Criteria
- Active Set Methods
- Feasible Starting Point
- Line-Search Methods
- Restricting the Step Length
- Computational Problems
- Covariance Matrix
- Input and Output Data Sets
- Displayed Output
- Missing Values
- Computational Resources
- Memory Limit
- Examples: NLP Procedure
- Example 4.1: Using the DATA= Option
- Example 4.2: Using the INQUAD= Option
- Example 4.3: Using the INEST=Option
- Example 4.4: Restarting an Optimization
- Example 4.5: Approximate Standard Errors
- Example 4.6: Maximum Likelihood Weibull Estimation
- Example 4.7: Simple Pooling Problem
- Example 4.8: Chemical Equilibrium
- Example 4.9: Minimize Total Delay in a Network
- References
- The NETFLOW Procedure
- Overview: NETFLOW Procedure
- Introduction
- Network Models
- Side Constraints
- Advantages of Network Models over LP Models
- Mathematical Description of NPSC
- Flow Conservation Constraints
- Nonarc Variables
- Warm Starts
- Getting Started: NETFLOW Procedure
- Syntax: NETFLOW Procedure
- Functional Summary
- Dictionary of Options
- Interactivity
- PROC NETFLOW Statement
- CAPACITY Statement
- COEF Statement
- COLUMN Statement
- CONOPT Statement
- COST Statement
- DEMAND Statement
- HEADNODE Statement
- ID Statement
- LO Statement
- MULT Statement
- NAME Statement
- NODE Statement
- PIVOT Statement
- PRINT Statement
- QUIT Statement
- RESET Statement
- RHS Statement
- ROW Statement
- RUN Statement
- SAVE Statement
- SHOW Statement
- SUPDEM Statement
- SUPPLY Statement
- TAILNODE Statement
- TYPE Statement
- VAR Statement
- Details: NETFLOW Procedure
- Input Data Sets
- Output Data Sets
- Converting Any PROC NETFLOW Format to an MPS-Format SAS Data Set
- Case Sensitivity
- Loop Arcs
- Multiple Arcs
- Pricing Strategies
- Dual Variables, Reduced Costs, and Status
- The Working Basis Matrix
- Flow and Value Bounds
- Tightening Bounds and Side Constraints
- Reasons for Infeasibility
- Missing S Supply and Missing D Demand Values
- Balancing Total Supply and Total Demand
- Warm Starts
- How to Make the Data Read of PROC NETFLOW More Efficient
- Macro Variable _ORNETFL
- Memory Limit
- The Interior Point Algorithm: NETFLOW Procedure
- Introduction
- Network Models: Interior Point Algorithm
- Linear Programming Models: Interior Point Algorithm
- Generalized Networks: NETFLOW Procedure
- Using the New EXCESS= Option in Pure Networks: NETFLOW Procedure
- Handling Excess Supply or Demand
- Handling Missing Supply and Demand Simultaneously
- Maximum Flow Problems
- Handling Supply and Demand Ranges
- Using the New EXCESS= Option in Generalized Networks: NETFLOW Procedure
- Total Supply and Total Demand: How Generalized Networks Differ from Pure Networks
- The EXCESS=SUPPLY Option
- The EXCESS=DEMAND Option
- Examples: NETFLOW Procedure
- Example 5.1: Shortest Path Problem
- Example 5.2: Minimum Cost Flow Problem
- Example 5.3: Using a Warm Start
- Example 5.4: Production, Inventory, Distribution Problem
- Example 5.5: Using an Unconstrained Solution Warm Start
- Example 5.6: Adding Side Constraints, Using a Warm Start
- Example 5.7: Using a Constrained Solution Warm Start
- Example 5.8: Nonarc Variables in the Side Constraints
- Example 5.9: Pure Networks: Using the EXCESS= Option
- Example 5.10: Maximum Flow Problem
- Example 5.11: Generalized Networks: Using the EXCESS= Option
- Example 5.12: Generalized Networks: Maximum Flow Problem
- Example 5.13: Machine Loading Problem
- Example 5.14: Generalized Networks: Distribution Problem
- Example 5.15: Converting to an MPS-Format SAS Data Set
- References
- The OPTMODEL Procedure
- Overview: OPTMODEL Procedure
- Getting Started: OPTMODEL Procedure
- OPTMODEL Modeling Language: Basic Concepts
- Named Parameters
- Indexing
- Types
- Names
- Parameters
- Expressions
- Identifier Expressions
- Function Expressions
- Index Sets
- Syntax: OPTMODEL Procedure
- Functional Summary
- PROC OPTMODEL Statement
- Declaration Statements
- CONSTRAINT Declaration
- MAX and MIN Objective Declarations
- NUMBER, STRING, and SET Parameter Declarations
- VAR Declaration
- Programming Statements
- Assignment Statement
- CALL Statement
- CLOSEFILE Statement
- CONTINUE Statement
- CREATE DATA Statement
- DO Statement
- DO Statement, Iterative
- DO UNTIL Statement
- DO WHILE Statement
- DROP Statement
- EXPAND Statement
- FILE Statement
- FIX Statement
- FOR Statement
- IF Statement
- LEAVE Statement
- Null Statement
- PRINT Statement
- PUT Statement
- READ DATA Statement
- RESET OPTIONS Statement
- RESTORE Statement
- SAVE MPS Statement
- SAVE QPS Statement
- SOLVE Statement
- STOP Statement
- UNFIX Statement
- Macro Variable _OROPTMODEL_
- OPTMODEL Expression Extensions
- AND Aggregation Expression
- CARD Function
- CROSS Expression
- DIFF Expression
- IF-THEN/ELSE Expression
- IN Expression
- Index Set Expression
- INTER Expression
- INTER Aggregation Expression
- MAX Aggregation Expression
- MIN Aggregation Expression
- OR Aggregation Expression
- PROD Aggregation Expression
- Range Expression
- Set Constructor Expression
- Set Literal Expression
- SETOF Aggregation Expression
- SLICE Expression
- SUM Aggregation Expression
- SYMDIFF Expression
- Tuple Expression
- UNION Expression
- UNION Aggregation Expression
- WITHIN Expression
- Details: OPTMODEL Procedure
- Conditions of Optimality
- Data Set Input/Output
- Control Flow
- Formatted Output
- ODS Table and Variable Names
- Constraints
- Suffixes
- Integer Variable Suffixes
- Dual Values
- Reduced Costs
- Presolver
- Model Update
- OPTMODEL Options
- Automatic Differentiation
- Conversions
- More on Index Sets
- Memory Limit
- Examples: OPTMODEL Procedure
- Example 6.1: Matrix Square Root
- Example 6.2: Reading from and Creating a Data Set
- Example 6.3: Model Construction
- Example 6.4: Set Manipulation
- Rewriting NLP Models for PROC OPTMODEL
- References
- The Interior Point Nonlinear Programming Solver -- Experimental
- Overview
- Getting Started
- Syntax
- Details
- Basic Definitions and Notation
- Overview of Constrained Optimization
- Overview of Interior Point Methods
- Solver Termination Criterion
- Solver Termination Messages
- Macro Variable _OROPTMODEL_
- Examples
- Example 7.1: Solving Highly Nonlinear Optimization Problems
- Example 7.2: Solving Unconstrained Optimization Problems
- Example 7.3: Solving NLP Problems with Range Constraints
- References
- The Linear Programming Solver
- Overview
- Getting Started
- Syntax
- Details
- Presolve
- Pricing Strategies for the Primal Simplex Solver
- The Interior Point Algorithm: Overview
- Macro Variable _OROPTMODEL_
- Iteration Log for the Simplex Solvers
- Iteration Log for the Interior Point Solver
- Problem Statistics
- Data Magnitude and Variable Bounds
- Variable and Constraint Status
- Irreducible Infeasible Set
- Examples
- Example 8.1: Diet Problem
- Example 8.2: Reoptimizing the Diet Problem Using BASIS=WARMSTART
- Example 8.3: Two-Person Zero-Sum Game
- Example 8.4: Finding an Irreducible Infeasible Set
- References
- The Mixed Integer Linear Programming Solver
- Overview
- Getting Started
- Syntax
- Details
- The Branch-and-Bound Algorithm
- Controlling the Branch-and-Bound Algorithm
- Presolve and Probing
- Cutting Planes
- Primal Heuristics
- Node Log
- Problem Statistics
- Data Magnitude and Variable Bounds
- Macro Variable _OROPTMODEL_
- Examples
- Example 9.1: Scheduling
- Example 9.2: Multicommodity Transshipment Problem with Fixed Charges
- Example 9.3: Facility Location
- Example 9.4: Traveling Salesman Problem
- References
- The NLPC Nonlinear Optimization Solver
- Overview
- Getting Started
- Syntax
- Details
- Optimization Algorithms
- Conditions of Optimality
- Optimality Control
- Infeasibility
- Feasible Starting Point
- Line-Search Method
- Computational Problems
- Iteration Log
- Macro Variable _OROPTMODEL_
- Examples
- Example 10.1: Least-Squares Problem
- Example 10.2: Maximum Likelihood Weibull Model
- Example 10.3: Simple Pooling Problem
- References
- The Unconstrained Nonlinear Programming Solver
- Overview
- Getting Started
- Syntax
- Details
- Conditions of Optimality
- Line-Search Algorithm
- Broyden-Fletcher-Goldfarb-Shanno (BFGS) Algorithm
- Macro Variable _OROPTMODEL_
- Examples
- Example 11.1: Solving a Highly Nonlinear Problem
- Example 11.2: Solving the Accumulated Rosenbrock Function
- References
- The Quadratic Programming Solver -- Experimental
- Overview
- Getting Started
- Syntax
- Details
- Examples
- Example 12.1: Linear Least-Squares Problem
- Example 12.2: Portfolio Optimization
- Example 12.3: Portfolio Selection with Transactions
- References
- The Sequential Quadratic Programming Solver
- Overview
- Getting Started
- Syntax
- Details
- Examples
- Example 13.1: Solving a Highly Nonlinear Problem
- Example 13.2: Using the HESCHECK Option
- Example 13.3: Choosing a Good Starting Point
- Example 13.4: Using the PENALTY= Option
- Example 13.5: Unconstrained NLP Optimization
- Example 13.6: Solving Large-Scale NLP Problems
- References
- The MPS-Format SAS Data Set
- Overview
- Sections Format
- NAME Section
- ROWS Section
- COLUMNS Section
- RHS Section (Optional)
- RANGES Section (Optional)
- BOUNDS Section (Optional)
- BRANCH Section (Optional)
- QSECTION Section
- ENDATA Section
- Details
- Examples
- Example 14.1: MPS-Format Data Set for a Product Mix Problem
- Example 14.2: Fixed-MPS-Format File
- Example 14.3: Free-MPS-Format File
- Example 14.4: Using the %MPS2SASD Macro
- References
- The OPTLP Procedure
- Overview: OPTLP Procedure
- Getting Started: OPTLP Procedure
- Syntax: OPTLP Procedure
- Details: OPTLP Procedure
- Data Input and Output
- Presolve
- Pricing Strategies for the Simplex Solvers
- Warm Start for the Simplex Solvers
- The Interior Point Algorithm: Overview
- Iteration Log for the Simplex Solvers
- Iteration Log for the Interior Point Solver
- ODS Tables
- Irreducible Infeasible Set
- Memory Limit
- Examples: OPTLP Procedure
- Example 15.1: Oil Refinery Problem
- Example 15.2: Using the Interior Point Solver
- Example 15.3: The Diet Problem
- Example 15.4: Reoptimizing after Modifying the Objective Function
- Example 15.5: Reoptimizing after Modifying the Right-Hand Side
- Example 15.6: Reoptimizing after Adding a New Constraint
- Example 15.7: Finding an Irreducible Infeasible Set
- References
- The OPTMILP Procedure
- Overview: OPTMILP Procedure
- Getting Started: OPTMILP Procedure
- Syntax: OPTMILP Procedure
- Details: OPTMILP Procedure
- Data Input and Output
- Warm Start
- The Branch-and-Bound Algorithm
- Controlling the Branch-and-Bound Algorithm
- Presolve and Probing
- Cutting Planes
- Primal Heuristics
- Node Log
- ODS Tables
- Memory Limit
- Examples: OPTMILP Procedure
- Example 16.1: Simple Integer Linear Program
- Example 16.2: MIPLIB Benchmark Instance
- Example 16.3: Facility Location
- References
- The OPTQP Procedure
- Overview: OPTQP Procedure
- Getting Started: OPTQP Procedure
- Syntax: OPTQP Procedure
- Details: OPTQP Procedure
- Output Data Sets
- Interior Point Algorithm: Overview
- Iteration Log for the OPTQP Procedure
- ODS Tables
- Memory Limit
- Examples: OPTQP Procedure
- Example 17.1: Linear Least-Squares Problem
- Example 17.2: Portfolio Optimization
- Example 17.3: Portfolio Selection with Transactions
- References
Index here
| Product | Release |
|---|---|
| SAS/OR | 9.2 |
- Type
- Usage and Reference
- Copyright Date
- March 2008
- Last Updated
- 24Feb2009
| The INTPOINT Procedure |
Mathematical Description of LP
A linear programming (LP) problem has a linear objective function and a collection of linear constraints. PROC INTPOINT finds the values of variables that minimize the total cost of the solution. The value of each variable is on or between the variable's lower and upper bounds, and the constraints are satisfied.
If an LP has 
variables and 
constraints, then
the formal statement of the problem solved by PROC INTPOINT is
where
-
is the 
variable objective function coefficient vector
-
is the variable value vector
-
is the 
constraint coefficient matrix for
the variables, where
is the coefficient of variable 
in the 
th
constraint
-
is the 
side constraint right-hand-side vector
-
is the variable lower bound vector
-
is the variable upper bound vector
Copyright © 2008 by SAS Institute Inc., Cary, NC, USA. All rights reserved.
