A symbolic reformulation/spatial branch-and-bound algorithm for the global optimisation of nonconvex MINLPs

Feasibility-Based Bounds Tightening via Fixed Points

“…The expression representation for functions is well known in computer science [15], engineering [16,17] and GO [18][19][20] where it is used for two substeps of the sBB algorithm: lower bound computation and FBBT. More formal constructions of E can be found in [21], Sect.…”

“…Some are based on the factorability of the functions in g(x) [25][26][27], whilst others are based on a symbolic reformulation of (1) based on the problem DAG D [18][19][20]. We employ the latter approach: each non-leaf node z in the vertex set of D is replaced by an added variable w z and a corresponding constraint (4) is adjoined to to the formulation (usually, two variables w v , w u corresponding to identical defining constraints are replaced by one single added variable).…”

“…We employ the latter approach: each non-leaf node z in the vertex set of D is replaced by an added variable w z and a corresponding constraint (4) is adjoined to to the formulation (usually, two variables w v , w u corresponding to identical defining constraints are replaced by one single added variable). The resulting reformulation, sometimes called Smith standard form [18], is exact [14]. A linear relaxation of (1) can automatically be obtained by the Smith standard form by replacing each nonlinear defining constraints by lower and upper linear approximations.…”

“…The constraints for U can be trivially reformulated from interval to scalar arithmetic; the equivalent reformulation for constraints in D is a little more involved but very well known [18,19,5]. The LP encoding the gfp of the FBBT is given in (14)- (34).…”

“…There are two commonly used range reduction techniques in Global Optimization (GO): Optimization Based Bounds Tightening (OBBT) and FBBT. The former is slower and more effective, involves the solution of 2n Linear Programs (LP) and is used either rarely or just at the root node of the sBB search tree [18]. The latter is an iterative procedure based on propagating the effect of the constraints g(x) ∈ G 0 on the variable ranges using interval arithmetic; FBBT is known to be practically efficient and is normally used at each sBB node.…”