Binary (or 0-1) knapsack problem

Given a knapsack of some capacity $C$ and $n$ objects with object $i$ having weight $w_i$ and profit $p_i$, the goal is to choose some subset of the objects that can fit in the knapsack (that is, the sum of their weights is no more than $C$) while maximizing profit.

This can be formulated as a mixed-integer program as:

\[\begin{array}{ll} \text{maximize} & x' p \\ \text{subject to} & x \in \{0, 1\} \\ & w' x \leq C \\ \end{array}\]

where $x$ is a vector is size $n$ where $x_i$ is one if we chose to keep the object in the knapsack, 0 otherwise.

# Data taken from http://people.sc.fsu.edu/~jburkardt/datasets/knapsack_01/knapsack_01.html
w = [23; 31; 29; 44; 53; 38; 63; 85; 89; 82]
C = 165
p = [92; 57; 49; 68; 60; 43; 67; 84; 87; 72];
n = length(w)
10
using Convex, GLPK
x = Variable(n, BinVar)
problem = maximize(dot(p, x), dot(w, x) <= C)
solve!(problem, GLPK.Optimizer)
Problem statistics
  problem is DCP         : true
  number of variables    : 1 (10 scalar elements)
  number of constraints  : 1 (1 scalar elements)
  number of coefficients : 21
  number of atoms        : 5

Solution summary
  termination status : OPTIMAL
  primal status      : FEASIBLE_POINT
  dual status        : NO_SOLUTION
  objective value    : 309.0

Expression graph
  maximize
   └─ sum (affine; real)
      └─ .* (affine; real)
         ├─ 10×1 Matrix{Int64}
         └─ 10-element real variable (id: 640…423)
  subject to
   └─ ≤ constraint (affine)
      └─ + (affine; real)
         ├─ sum (affine; real)
         │  └─ …
         └─ [-165;;]
evaluate(x)
10-element Vector{Float64}:
 1.0
 1.0
 1.0
 1.0
 0.0
 1.0
 0.0
 0.0
 0.0
 0.0

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