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1# ------------------------------------------------------------------------------ 

2# Copyright (C) 2012-2017 Guillaume Sagnol 

3# Copyright (C) 2018-2019 Maximilian Stahlberg 

4# 

5# This file is part of PICOS. 

6# 

7# PICOS is free software: you can redistribute it and/or modify it under the 

8# terms of the GNU General Public License as published by the Free Software 

9# Foundation, either version 3 of the License, or (at your option) any later 

10# version. 

11# 

12# PICOS is distributed in the hope that it will be useful, but WITHOUT ANY 

13# WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR 

14# A PARTICULAR PURPOSE. See the GNU General Public License for more details. 

15# 

16# You should have received a copy of the GNU General Public License along with 

17# this program. If not, see <http://www.gnu.org/licenses/>. 

18# ------------------------------------------------------------------------------ 

19 

20"""Implementation of :class:`SumExtremesConstraint`.""" 

21 

22from collections import namedtuple 

23 

24import cvxopt as cvx 

25 

26from .. import glyphs 

27from ..apidoc import api_end, api_start 

28from .constraint import Constraint, ConstraintConversion 

29 

30_API_START = api_start(globals()) 

31# ------------------------------- 

32 

33 

34class SumExtremesConstraint(Constraint): 

35 """Bound on a sum over extreme (eigen)values.""" 

36 

37 class Conversion(ConstraintConversion): 

38 """Sum over extremes to LMI/affine constraint conversion.""" 

39 

40 @classmethod 

41 def predict(cls, subtype, options): 

42 """Implement :meth:`~.constraint.ConstraintConversion.predict`.""" 

43 from ..expressions import (HermitianVariable, RealVariable, 

44 SymmetricVariable) 

45 from . import AffineConstraint, ComplexLMIConstraint, LMIConstraint 

46 

47 nm, k, eigenvalues, complex = subtype 

48 

49 # Determine matrix variable dimension. 

50 if eigenvalues: 

51 nFloat = nm**0.5 

52 n = int(nFloat) 

53 assert n == nFloat 

54 d = n**2 if complex else n*(n + 1) // 2 

55 else: 

56 n = nm 

57 

58 # Validate k. 

59 assert k > 0 and k <= n 

60 

61 # Define shorthands for better formatting below. 

62 RV, AC = RealVariable, AffineConstraint 

63 LMI = ComplexLMIConstraint if complex else LMIConstraint 

64 MV = HermitianVariable if complex else SymmetricVariable 

65 

66 if eigenvalues: 

67 if k == 1: 

68 yield ("con", LMI.make_type(diag=n), 1) 

69 elif k == n: 

70 # NOTE: Refinement prevents this case from happening. 

71 yield ("con", AC.make_type(dim=1, eq=False), 1) 

72 else: 

73 yield ("var", RV.make_var_type(dim=1, bnd=0), 1) 

74 yield ("var", MV.make_var_type(dim=d, bnd=0), 1) 

75 yield ("con", LMI.make_type(diag=n), 2) 

76 yield ("con", AC.make_type(dim=1, eq=False), 1) 

77 else: 

78 if k == 1: 

79 yield ("con", AC.make_type(dim=n, eq=False), 1) 

80 elif k == n: 

81 # NOTE: Refinement prevents this case from happening. 

82 yield ("con", AC.make_type(dim=1, eq=False), 1) 

83 else: 

84 yield ("var", RV.make_var_type(dim=1, bnd=0), 1) 

85 yield ("var", RV.make_var_type(dim=n, bnd=n), 1) 

86 yield ("con", AC.make_type(dim=n, eq=False), 1) 

87 yield ("con", AC.make_type(dim=1, eq=False), 1) 

88 

89 @classmethod 

90 def convert(cls, con, options): 

91 """Implement :meth:`~.constraint.ConstraintConversion.convert`.""" 

92 from ..expressions import (Constant, HermitianVariable, 

93 RealVariable, SymmetricVariable) 

94 from ..modeling import Problem 

95 

96 theSum = con.theSum 

97 relation = con.relation 

98 rhs = con.rhs 

99 

100 x = theSum.x 

101 k = theSum.k 

102 

103 if theSum.eigenvalues: 

104 n = x.shape[0] 

105 I = Constant('I', cvx.spdiag([1.] * n)) 

106 else: 

107 n = len(x) 

108 

109 if x.complex: 

110 MatrixVariable = HermitianVariable 

111 else: 

112 MatrixVariable = SymmetricVariable 

113 

114 P = Problem() 

115 

116 if relation == Constraint.LE: 

117 if theSum.eigenvalues: 

118 if k == 1: 

119 P.add_constraint(x << rhs * I) 

120 elif k == n: 

121 # NOTE: Refinement prevents this case from happening. 

122 P.add_constraint(("I" | x) <= rhs) 

123 else: 

124 s = RealVariable('s') 

125 Z = MatrixVariable('Z', n) 

126 P.add_constraint(Z >> 0) 

127 P.add_constraint(x << Z + s * I) 

128 P.add_constraint(rhs >= (I | Z) + (k * s)) 

129 else: 

130 if k == 1: 

131 P.add_constraint(x <= rhs) 

132 elif k == n: 

133 P.add_constraint((1 | x) <= rhs) 

134 else: 

135 lbda = RealVariable('lambda') 

136 mu = RealVariable('mu', x.shape, lower=0) 

137 P.add_constraint(x <= lbda + mu) 

138 P.add_constraint(k * lbda + (1 | mu) <= rhs) 

139 else: 

140 if theSum.eigenvalues: 

141 if k == 1: 

142 P.add_constraint(x >> rhs * I) 

143 elif k == n: 

144 # NOTE: Refinement prevents this case from happening. 

145 P.add_constraint((I | x) <= rhs) 

146 else: 

147 s = RealVariable('s') 

148 Z = MatrixVariable('Z', n) 

149 P.add_constraint(Z >> 0) 

150 P.add_constraint(-x << Z + s * I) 

151 P.add_constraint(-rhs >= (I | Z) + (k * s)) 

152 else: 

153 if k == 1: 

154 P.add_constraint(x >= rhs) 

155 elif k == n: 

156 P.add_constraint((1 | x) >= rhs) 

157 else: 

158 lbda = RealVariable('lambda') 

159 mu = RealVariable('mu', x.shape, lower=0) 

160 P.add_constraint(-x <= lbda + mu) 

161 P.add_constraint(k * lbda + (1 | mu) <= -rhs) 

162 

163 return P 

164 

165 def __init__(self, theSum, relation, rhs): 

166 """Construct a :class:`SumExtremesConstraint`. 

167 

168 :param ~picos.expressions.SumExtremes theSum: 

169 Left hand side expression. 

170 :param str relation: 

171 Constraint relation symbol. 

172 :param ~picos.expressions.AffineExpression rhs: 

173 Right hand side expression. 

174 """ 

175 from ..expressions import AffineExpression, SumExtremes 

176 

177 assert isinstance(theSum, SumExtremes) 

178 assert isinstance(rhs, AffineExpression) 

179 assert relation in self.LE + self.GE 

180 assert len(rhs) == 1 

181 

182 self.theSum = theSum 

183 self.relation = relation 

184 self.rhs = rhs 

185 

186 super(SumExtremesConstraint, self).__init__(theSum._typeStr) 

187 

188 Subtype = namedtuple("Subtype", ("argdim", "k", "eigenvalues", "complex")) 

189 

190 def _subtype(self): 

191 return self.Subtype(len(self.theSum.x), self.theSum.k, 

192 self.theSum.eigenvalues, self.theSum.x.complex) 

193 

194 @classmethod 

195 def _cost(cls, subtype): 

196 nm, _, eigenvalues, _ = subtype 

197 

198 if eigenvalues: 

199 nFloat = nm**0.5 

200 n = int(nFloat) 

201 assert n == nFloat 

202 else: 

203 n = nm 

204 

205 return n + 1 

206 

207 def _expression_names(self): 

208 yield "theSum" 

209 yield "rhs" 

210 

211 def _str(self): 

212 if self.relation == self.LE: 

213 return glyphs.le(self.theSum.string, self.rhs.string) 

214 else: 

215 return glyphs.ge(self.theSum.string, self.rhs.string) 

216 

217 def _get_slack(self): 

218 if self.relation == self.LE: 

219 return self.rhs.safe_value - self.theSum.safe_value 

220 else: 

221 return self.theSum.safe_value - self.rhs.safe_value 

222 

223 

224# -------------------------------------- 

225__all__ = api_end(_API_START, globals())