Separable Programming: Theory and MethodsIn this book, the author considers separable programming and, in particular, one of its important cases - convex separable programming. Some general results are presented, techniques of approximating the separable problem by linear programming and dynamic programming are considered. Convex separable programs subject to inequality/ equality constraint(s) and bounds on variables are also studied and iterative algorithms of polynomial complexity are proposed. As an application, these algorithms are used in the implementation of stochastic quasigradient methods to some separable stochastic programs. Numerical approximation with respect to I1 and I4 norms, as a convex separable nonsmooth unconstrained minimization problem, is considered as well. Audience: Advanced undergraduate and graduate students, mathematical programming/ operations research specialists. |
Contents
IV | 1 |
V | 8 |
VI | 15 |
VII | 24 |
VIII | 41 |
IX | 53 |
X | 62 |
XI | 63 |
XLVIII | 182 |
XLIX | 184 |
L | 187 |
LI | 191 |
LII | 193 |
LIII | 195 |
LIV | 199 |
LV | 207 |
XIII | 65 |
XIV | 79 |
XV | 91 |
XVI | 98 |
XVII | 106 |
XVIII | 108 |
XIX | 111 |
XX | 112 |
XXI | 116 |
XXII | 120 |
XXIII | 122 |
XXIV | 128 |
XXV | 134 |
XXVI | 137 |
XXVII | 140 |
XXVIII | 141 |
XXXII | 143 |
XXXIII | 151 |
XXXIV | 154 |
XXXV | 159 |
XXXVI | 162 |
XXXVII | 163 |
XXXVIII | 168 |
XXXIX | 170 |
XL | 173 |
XLI | 175 |
XLIV | 179 |
XLV | 181 |
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Common terms and phrases
according to Theorem Algorithm assumption best approximation Chapter Consider convergence convex combination convex function convex programming convex separable convex set Corollary defined Definition Denote dynamic programming epi ƒ equality constraint exists f(xk f(xo feasible region feasible solution follows function f ƒ is convex gi(x go to Step grid points Hence hyperplane I₁ inf f(x inventory J₁ JA(k JEJA JEJA JEJA(K KKT conditions Lagrangian LASP Let f Let X1 linear lower semicontinuous method minimization minimum monotone nonincreasing nonempty nonnegative normed objective function obtain optimal solution polytope problem SP real number respectively saddle point satisfied semicontinuous separable problem separable programming sequence solution to problem solving stochastic quasigradient Subdifferential subgradient subgradient method Theorem 5.1 variables vector vili(x Xk+1