# 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: 178…933)
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