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Quadratic Knapsack Problem

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Quadratic Knapsack Problem

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  1. Knapsack Problem The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total weight is less than or equal to a given limit and the total value is as large as possible.

  2. BIG TEXT Example of a one-dimensional (constraint) knapsack problem: which boxes should be chosen to maximize the amount of money while still keeping the overall weight under or equal to 15 kg? A multiple constrained problem could consider both the weight and volume of the boxes. (Solution: if any number of each box is available, then three yellow boxes and three grey boxes; if only the shown boxes are available, then all except for the green box.)

  3. Types of Knapsack Problems 1. Multi-objective knapsack problem 2. Multiple knapsack problem 3. Quadratic knapsack problem 4. Subset-sum problem 5. Geometric knapsack problem

  4. Quadratic Knapsack Problem The quadratic knapsack problem (QKP), first introduced in 19th century,[1] is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a weight, a value, and an extra profit that can be earned if two items are selected, determine the number of items to include in a collection without exceeding capacity of the knapsack, so as to maximize the overall profit. Usually, quadratic knapsack problems come with a restriction on the number of copies of each kind of item: either 0, or 1. The quadratic knapsack problem maximizes a quadratic objective function subject to binary and linear capacity constraints. The problem was introduced by Gallo, Hammer, and Simeone in 1980, however the first treatment of the problem dates back to Witzgall in 1975.

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