Index. affine dependency, 133 minimal, 133 affine hull, 392 affinely independent, 393 α-bb, 230, 258, 297 approximate solutions, 9 approximation, 86
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1 Index affine dependency, 133 minimal, 133 affine hull, 392 affinely independent, 393 α-bb, 230, 258, 297 approximate solutions, 9 approximation, 86 AP X, 10 aspiration level, 86, 110 atomic clusters, 372 augmented Lagrangian, 266 backtracking, 49 BARON, 335 basin hopping, 45, 56, 75, 81, 376, 388 monotonic, 75 Bayesian approach, 117 best start, 51 best-first, 292 bias, 102 billiard simulation, 384 binary clusters, 378 bisection rule, 298 Boltzmann distribution, 75 Boolean quadric polytopes, 205 bound factors, 163 bound improving selection, 293 branch and bound (BB), 289 algorithm, 290, 382 tree, 291 branching, 295 breadth-first, 292 bumpiness, 87, 93 Carathéodory s theorem, 11, 392 CEC test problems, 369 centered forms, 257 circuit intersection point, 134 clique, 23 clique inequalities, 205 clustering methods, 52 clusters, 372 combination rule child-parent, 49 child-population, 49 child-population Gauss Seidel, 49 completely monotonic function, 94 completely positive matrices, 395 concave envelope, 127 concave function, 395 concave minimization, 259, 309, 350 concentrated log likelihood, 105 cone, 309 completely positive, 26 copositive, 26 polyhedral, 309 conformational space annealing, 62, 376 conic programming, 26 conjugate function, 128 constraint filtering scheme, 170 constraint propagation, 348 constraint qualification, 260 continuation methods, 83 convex envelope, 79, 127 convex extension property, 281 convex function, 395 convex hull, 391 convex outer approximation, 126, 275 convex relaxation, 125 convex set, 391 convex underestimator, 126 cooling schedule, 75 copositive bound, 248 copositive cone, 26, 302 copositive matrices, 395 critical value,
2 434 Index cut-type inequalities, 205 cutting planes, 335 decision problem, 9 decomposition difference-of-convex (DC), 240 deep space maneuver, 386 δ-feasible, 289 δ-optimal solution, 290 depth-first, 292 deterministic sequential algorithm, 121 difference of convex (DC), 240 canonical form, 242 decomposition, 240, 321 spectral, 246 function, 240 optimization problem, 242, 315 undominated decomposition, 243, 244 difference of monotonic (DM) function, 251 problem, 324 differential evolution (DE), 66, 388 DIRECT, 266 direct, 69 direct move, 59 direct mutation, 377 dissimilarity, 61 criterion, 376 measure, 61 distance geometry, 84, 378 domain reduction, 153, 335 feasibility based, 335 optimality based, 335 doubly nonnegative matrices, 395 dual bounds, 259 dual cone, 394 dynamic lattice, 377 eccentricity, 305 edge-concave, 132 ellipsoid, 321 epigraph, 273, 396 ε-approximate solution, 10 ε-optimal solution, 290 (ε,δ)-optimal solution, 290 essential ε-optimal solution, 334 essential feasible set, 334 exact penalty, 266 exactness in the limit, 294 exhaustiveness property, 296 expected improvement, 117 exponential-type augmented Lagrangian, 266 extreme face, 350 extreme point, 392 facet representation, 158 facial cut, 351 factorable function, 267 fathoming rules, 358 feasibility based domain reduction, 335 feasible region, 3 filled function, 77 filter method, 388 force fields, 372 Frobenius inner product, 26, 394 fully polynomial time approximation scheme (FPTAS), 10 funnel, 47, 75, 370 bottom, 47 hopping, 377 γ -concavity cut, 314, 350 γ -extension, 313, 352 γ -valid cut, 352 generalized critical value, 225 generalized trust region problem, 14 generating set, 129 genetic algorithms, 66 geometric branching, 295 Gerschgorin theorem, 231 Gibbs distribution, 75 global information, 121 global minima, 3 global phase, 41 GLOBALlib, 370 GloballyGenerate, 44 gradient ideal, 220 greedy randomized adaptive search (GRASP), 68 hidden convexity, 12 hit-and-run, 79 homotopy methods, 83
3 Index 435 ideal, 219 improvement, 41 strict, 41 inclusion function, 256, 358 increasing function, 250 interpolation, 86 multivariate, 88 polynomial, 89 radial function, 90 intersection cut, 314 interval analysis, 358 interval arithmetic, 231, 255, 297 isolated singularities at infinity, 226 isotone, 256 isotonic property, 326 iterated local search, 45, 56, 57 Karush Kuhn Tucker (KKT), 23 conditions, 23, 325 point, 23 Kriging, 101, 117 Kuhn s triangulation, 137, 148 Lagrange function, 96 Lagrangian dual, 260 function, 259 Laplacian matrix, 29 LeGO, 85 Lennard-Jones, 372 linear (sum-of-ratios) fractional programming, 18 linear boundary value form, 258 linear complementarity, 286 linear constraint factors, 165 linear fractional multiplicative programming problem, 17 linear multiplicative programming, 18 linearity domains, 158 linearization phase, 163 Lipschitz constant, 252 Lipschitz function, 252 local information, 120 local Lipschitz, 255 local optimum, 3, 45 at level 0, 46 at level 1, 46 local phase, 41 local smoothing, 81 localizes, 122 LocallyGenerate, 44 Lovász extension, 138 Markov stochastic processes, 75 max bisection, 39 max cut, 29 maximum clique problem, 23, 26 mean value form function, 257 memetic algorithms, 48 merit function, 86, 87 model of computation bit model, 8 black box, 9 molecular conformation, 84, 371 moment, 227 monotonic basin hopping (MBH), 57, 75 monotonic optimization, 250 monotonicity test, 358 Morse potential, 72, 372 multilevel single linkage, 53 multilinear functions, 161, 286 Multistart, 51, 52 multivariate interpolation, 88 n-simplex, 133 natural interval extension, 257 negative definite, 393 negative semidefinite, 393 nested sequence, 296 nonlinearities removal range reduction, 340 nonnegative matrices, 394 nonnegative polynomials, 218 normal conical algorithms, 319 normal set, 251 objective function, 3 ω-subdivision rule, 299 one row relaxation, 188 optimality based domain reduction, 335 order relation, 41 pair potential, 372 parallelotope, 37 Pareto, 70 particle swarm optimization (PSO), 63, 66
4 436 Index Peano curve, 254 Piyavskii Shubert algorithm, 253 planet swing-by, 386 polyblocks, 252 polyhedral convex envelope, 130 polyhedral subdivision, 155 polyhedron, 391 polynomial constraint factors, 166 polynomial programming, 217 polynomial time approximation scheme (PTAS), 10 polytope, 391 pooling problems, 370 population basin hopping (PBH), 59, 68 dissimilarity based, 61 embarassing parallelism, 60 greedy, 60 population-based methods, 42 positive definite, 393 positive semidefinite, 393 positively homogeneous function, 276 posynomial functions, 286 potential, 372 principal gradient tentacle, 225 projective error, 272 pure adaptive search (PAS), 78 pure random search (PRS), 50, 74, 78 quadratic programming (QP), 8, 16, 172 standard, 23 quadratically constrained quadratic programming (QCQP), 211 quasi-concave function, 11, 359, 397 quasi-convex function, 397 quotient ring, 219 radial basis center, 90 cubic, 91 function, 91 Gaussian, 91 linear, 91 multiquadric, 91 thin plate spline, 91 radial function, 90 radical, 219 radical ideal, 219 randomized algorithms, 9 range reduction, 335 nonlinearities removal, 340 standard, 339 ray, 309 real variety, 219 recession cone, 277, 393 reformulation phase, 163 reformulation-linearizationtechnique (RLT), 163 regression, 86, 106 regularization parameter, 108 relaxation, 125 restricted candidate list, 68 reverse convex constraint, 242, 314 reverse normal set, 251 reverse polyblock, 252, 324 robust, 388 sandwich algorithm, 272 scatter search, 84 Schur complement, 394 second-order cone, 395 sees the global optimum, 122 selection operator, 48 semidefinite program, 12 sensor localization, 379 separable quadratic knapsack problem, 16 sequential methods, 42 Shepard s interpolation, 369 Shor relaxation, 247 signomial, 286 simple linkage, 55 simplex, 11, 302, 391 simulated annealing, 48, 58, 74 Slater s condition, 260 smoothing methods, 79 space trajectory, 370, 386 space-filling curve, 254 spectral DC decomposition, 246 spline, 90 cubic, 98 cubic natural, 99 standard quadratic program (StQP), 23, 247 standard range reduction, 339 stochastic sequential algorithm, 122 strictly concave function, 395
5 Index 437 strictly conditional positive definite function, 93 strictly convex function, 395 strictly positive definite function, 93 strong consistency, 331 subgradient, 396 subgradient projection, 322 submodular, 137 subset problem, 16 sufficiently rich class of problems, 119 sum-of-squares (SOS), 218 relaxation, 221 support vector machine (SVM), 85 supporting hyperplane, 392 surrogate bound, 259 surrogate model, 85 surrogate-based optimization one stage methods, 86 two stage methods, 86 survival, 49 systematic error, 102 vertex polyhedral convex envelope, 130 vertical error, 272 weakly convex, 320 zero-dimensional, 219 tabu-search, 48 Taylor form function, 258 temperature, 75 test problems, 363 CEC, 369 Shepard, 369 trace, 394 triangle inequalities, 205 triangulation, 133 Kuhn, 137, 148 truncated tangency variety, 226 trust region problem, 12, 321 tunneling function, 76 method, 76 2LMP problem, 17 two-phase, 376 unisolvent, 90 unit simplex, 391 universal Kriging, 105 variable neighborhood search (VNS), 45, 63 variety, 219 vertex, 392
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