## Unconstrained 0-1 optimization and Lagrangean relaxation.(English)Zbl 0725.90067

The unconstrained 0-1 polynomial programming problem (PPP), (P1) is considered. An equivalent problem with nonnegative coefficients of the terms of degree 2 and more is used: $(P2)\quad Z_{P_ 2}=\max \{f(x,\bar x)| x\in \{0,1\}^ n,\quad \bar x\in \{0,1\}^{\{I_ c\}}=$
$\sum^{n}_{i=1}\ell_ ix_ i+\sum_{k\in P}d_ k\prod_{i\in Q(k)}x_ i+\sum_{k\in N}c_ k\bar x_{T(k)}\prod_{i\in R(k)}x_ i,$ (here $$\bar x_ i=1-x_ i)$$. A linear function $$p(x)$$, a roof for (P2), is constructed as an upper bounding function of $$f(x,\bar x)$$. Let R be the set of all roofs, then the dual roof is defined as $W(R)=\min_{p(x)\in R}\{\max_{x\in \{0,1\}^ n}p(x)\}$ and this value may be calculated by solving the LP problem $Z_{LP}=\max [\sum^{n}_{i=1}\ell_ ix_ i+\sum_{k\in P}d_ xt_ k+\sum_{k\in N}c_ kw_ k]$ subject to linear constraints. W(R) is an upper bound on $$Z_{P_ 1}$$ with the so-called roof duality gap $W(R)-Z_{P_ 1}=\min_{p(x)\in R}\{\max_{x\in \{0,1\}^ n}p(x)\}-\max_{x\in \{0,1\}^ n}\{\min_{p(x)\in R}p(x)\}.$ The problem (P3) is built from (P2) by substitution $$y_ i=\bar x_ i$$. A Lagrangean dual function LD($$\pi$$) is constructed for problem $$(P3)\quad Z_{LD}=\min_{(\pi)} LD(\pi).$$
One of the results is the following Theorem 1. $$W(R)=Z_{LD}.$$
It is shown that checking the existence of a roof duality gap is equivalent to the checking of the consistency of a 0-1 quadratic posiform. The results of the paper are illustrated on an example of a 0-1 cubic programming problem with 4 variables.

### MSC:

 90C09 Boolean programming
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### References:

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