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Intro to Integer-valued constrained optimization

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• An integer-valued constrained optimization problem derived from an real-valued LP
  • Real-valued constrained optimization: (linear program (LP) )

    Max: f(x1, x2, ..., xn) s.t.: g1(x1, x2, ..., xn) ≤ c1 g2(x1, x2, ..., xn) ≤ c1 .... x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0 Note: x1, x1, ... are "real" valued

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  • Integer-valued constrained optimization (integer program (IP) ) derived from the above LP is:

    Max: f(x1, x2, ..., xn) s.t.: g1(x1, x2, ..., xn) ≤ c1 g2(x1, x2, ..., xn) ≤ c1 .... x1 ≥ 0, x2 ≥ 0, ..., xn ≥ 0 x1, x2, ..., xn integer

    In other words:

    The solution of an Integer Program (IP) must be integers

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• The feasible region of an Integer Program
  • Effect of Integer constraints:

    The integer constraints constitute additional constraints (= limitation) on a feasible solution The effect is "shrinking" the feasible region

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  • Example:

    Linear Program	Corresponding Integer Program

    The feasible regions of the Linear Program (LP) and Integer Program (IP) are shown in the following figure:

    Explanation:

    The feasible region defined by the constraints of the Linear Program is the entire shaded region The integer program imposes more constraints Therefore: A subset of the feasible region of the LP will statify the constraints of the integer program Specifically: The "integral" points inside the feasible (shaded) region satisfy the integer value constraints Feasible "region" of the Integer Program: All the dots inside the shaded region form the feasible region for the Integer Program.

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• Relationship between a LP and its derived IP
  • Fact 1:

    Optimal solution of IP   ≤   Optimal solution of LP

    Reason:

    We know that: A solution x1, x2, ... xn that satisfies the constraints of the IP problem will also satisfy the constraints of the LP problem Therefore: The optimal solution x1opt, x2opt, ... xnopt of the IP problem is a solution of the LP problem In other words: OptimalSolution( IP ) ≤ OptimalSolution ( LP )

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  • Fact 2:

    Optimal solution of LP may not be equal to the optimal solution of the derived IP

    Example:

    Consider: The optimal solution of the Linear Program (LP) is: x1 = 2.25 (not integer) x2 = 3.75 (not integer) This solution is not feasible for the IP problem Therefore: Optimal Solution of IP may not be equal to the Optimal Solotion of LP

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  • Fact 3:

    IF the optimal solution of LP is integral (= all values are integers), then this solution is also the optimal solution of the derived IP

    Reason:

    This fact follows directly from Fact 1: OptimalSolution( IP ) ≤ OptimalSolution ( LP ) Because the OptimalSolution ( LP ) is integral (= all integers), the solution is feasible for the Integer Program (IP) and therefore: OptimalSolution( IP ) ≤ OptimalSolution ( LP ) becomes: OptimalSolution( IP ) = OptimalSolution ( LP )

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  • Fact 4:

    IF the Linear Program LP is infeasible (i.e., the feasible region is empty), then the Integer Program IP is also infeasible

    Reason:

    The feasible region of the IP is a subset of the feasible region of the LP So if the feasible region of the LP is empty (because the LP is infeasible), then the the feasible region of the IP is also empty Therefore: the IP is also infeasible

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• Complexity of Integer Programming
  • Integer Program is hard to solve:

    The Integer Program problem is a known NP-hard problem Wikipedia link: click here "NP-hard" problems (and the related "NP-complete" problems) are problems where: There are no known O(nk) algorithm for solving the problem (Algorithm to solve these problems have exponential order of running time (i.e., O(2n))

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  • Practice:

    Integer Programs are solved using heuristics

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• Heuristics to solve Integer Programming problems
  • Fact:

    An Integer Program is similar to its corresponding Linear Program (which can be solve efficiently)

    Idea:

    Let's try to use the corresponding Linear Program to solve the harder Integer Program

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  • Terminology:

    Linear relaxation of an Integer Program = the corresponding Linear Program obtained by dropping (omitting) the integer constraints from the Integer Program

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  • Commonly used heuristic:

    Branch and bound         Cutting planes

    We will study the branch and bound method next.