49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 class min_quad_with_fixed_precompute: def __init__(self, Q, A=None, k=None): """Prepare a precomputation object to efficiently solve the following constrained optimization problem: ``` argmin_u 0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u) A*u == b u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor) ``` Parameters ---------- Q : (n,n) symmetric sparse scipy csr_matrix This matrix will be symmetrized if not exactly symmetric. A : None or (m,n) sparse scipy csr_matrix m=0 assumed if None k : None or (o,) numpy array o=0 assumed if None Returns ------- precomputed : instance of class min_quad_with_fixed_precompute precomputation object that can be used to solve the optimization problem Examples -------- TODO """ self.n = Q.shape[0] assert Q.shape[1] == self.n assert self.n>0 # self.Q = Q.copy() if A is None: self.m = 0 self.A = None else: self.m = A.shape[0] assert A.shape[1] == self.n # self.A = A.copy() if k is None: self.o = 0 self.k = None else: self.o = k.shape[0] assert k.shape == (self.o,) assert np.min(k)>=0 and np.max(k)0 and c.shape==(self.n,p)) assert b is None or (np.isscalar(b) and self.m>0) or (p==0 and b.shape==(self.m,)) or (p>0 and b.shape==(self.m,p)) assert y is None or (np.isscalar(y) and self.o>0) or (p==0 and y.shape==(self.o,)) or (p>0 and y.shape==(self.o,p)) # Get everything to full dimensions if c is None: c = 0. if np.isscalar(c): if p==0: c = np.full(self.n, c) else: c = np.full((self.n,p), c) if b is None and self.m>0: b = 0. if np.isscalar(b) and self.m>0: if p==0: b = np.full(self.m, b) else: b = np.full((self.m,p), b) if y is None and self.o>0: y = 0. if np.isscalar(y) and self.o>0: if p==0: y = np.full(self.o, y) else: y = np.full((self.o,p), y) # Modified rhs based on known values if self.o==0: cmod = c else: c_reduced = c[self.ki] if p==0 else c[self.ki,:] cmod = c_reduced + self.Q_for_extra_c*y # We need to solve [Q, A.transpose(); A, 0] = [cmod; b] with the # reduced matrix Q. if self.m==0: rhs = -cmod else: # Modify b based on known values bmod = b if self.o==0 else (b - self.A_for_extra_b*y) rhs = np.concatenate([-cmod,bmod], axis=0) ured = self.solver(rhs) # Discard the dummy degrees of freedom from A*u == b ured = ured[0:self.n_reduced] if p==0 else ured[0:self.n_reduced,:] # Undo the u[k]==y reduction if self.o==0: u = ured else: if p==0: u = np.empty(self.n, dtype=np.float64) u[self.ki] = ured u[self.k] = y else: u = np.empty((self.n,p), dtype=np.float64) u[self.ki,:] = ured u[self.k,:] = y return u

__init__(Q, A=None, k=None)

Prepare a precomputation object to efficiently solve the following constrained optimization problem:

argmin_u  0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u)
A*u == b
u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor)

Parameters:

Name Type Description Default
Q (n,n) symmetric sparse scipy csr_matrix

This matrix will be symmetrized if not exactly symmetric.

required
A None or (m,n) sparse scipy csr_matrix

m=0 assumed if None

None
k None or (o,) numpy array

o=0 assumed if None

None

Returns:

Name Type Description

precomputation object that can be used to solve the optimization problem

Examples:

TODO

 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 def __init__(self, Q, A=None, k=None): """Prepare a precomputation object to efficiently solve the following constrained optimization problem: ``` argmin_u 0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u) A*u == b u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor) ``` Parameters ---------- Q : (n,n) symmetric sparse scipy csr_matrix This matrix will be symmetrized if not exactly symmetric. A : None or (m,n) sparse scipy csr_matrix m=0 assumed if None k : None or (o,) numpy array o=0 assumed if None Returns ------- precomputed : instance of class min_quad_with_fixed_precompute precomputation object that can be used to solve the optimization problem Examples -------- TODO """ self.n = Q.shape[0] assert Q.shape[1] == self.n assert self.n>0 # self.Q = Q.copy() if A is None: self.m = 0 self.A = None else: self.m = A.shape[0] assert A.shape[1] == self.n # self.A = A.copy() if k is None: self.o = 0 self.k = None else: self.o = k.shape[0] assert k.shape == (self.o,) assert np.min(k)>=0 and np.max(k)

solve(c=None, b=None, y=None)

Solve the following quadratic program with linear constraints:

argmin_u  0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u)
A*u == b
u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor)

Parameters:

Name Type Description Default
c None or scalar or (n,) numpy array or (n,p) numpy array

Assumed to be scalar 0 if None

None
b None or scalar or (m,) numpy array or (m,p) numpy array

Assumed to be scalar 0 if None

None
y None or scalar or (o,) numpy array or (o,p) numpy array

Assumed to be scalar 0 if None

None

Returns:

Name Type Description
u (n,) numpy array or (n,p) numpy array

Solution to the optimization problem

Examples:

TODO

 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 def solve(self, c=None, b=None, y=None): """Solve the following quadratic program with linear constraints: ``` argmin_u 0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u) A*u == b u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor) ``` Parameters ---------- c : None or scalar or (n,) numpy array or (n,p) numpy array Assumed to be scalar 0 if None b : None or scalar or (m,) numpy array or (m,p) numpy array Assumed to be scalar 0 if None y : None or scalar or (o,) numpy array or (o,p) numpy array Assumed to be scalar 0 if None Returns ------- u : (n,) numpy array or (n,p) numpy array Solution to the optimization problem Examples -------- TODO """ def cp(x): if x is None or np.isscalar(x): return 0 if len(x.shape)==1: return 0 return x.shape[1] p = max([cp(c), cp(b), cp(y)]) assert c is None or np.isscalar(c) or (p==0 and c.shape==(self.n,)) or (p>0 and c.shape==(self.n,p)) assert b is None or (np.isscalar(b) and self.m>0) or (p==0 and b.shape==(self.m,)) or (p>0 and b.shape==(self.m,p)) assert y is None or (np.isscalar(y) and self.o>0) or (p==0 and y.shape==(self.o,)) or (p>0 and y.shape==(self.o,p)) # Get everything to full dimensions if c is None: c = 0. if np.isscalar(c): if p==0: c = np.full(self.n, c) else: c = np.full((self.n,p), c) if b is None and self.m>0: b = 0. if np.isscalar(b) and self.m>0: if p==0: b = np.full(self.m, b) else: b = np.full((self.m,p), b) if y is None and self.o>0: y = 0. if np.isscalar(y) and self.o>0: if p==0: y = np.full(self.o, y) else: y = np.full((self.o,p), y) # Modified rhs based on known values if self.o==0: cmod = c else: c_reduced = c[self.ki] if p==0 else c[self.ki,:] cmod = c_reduced + self.Q_for_extra_c*y # We need to solve [Q, A.transpose(); A, 0] = [cmod; b] with the # reduced matrix Q. if self.m==0: rhs = -cmod else: # Modify b based on known values bmod = b if self.o==0 else (b - self.A_for_extra_b*y) rhs = np.concatenate([-cmod,bmod], axis=0) ured = self.solver(rhs) # Discard the dummy degrees of freedom from A*u == b ured = ured[0:self.n_reduced] if p==0 else ured[0:self.n_reduced,:] # Undo the u[k]==y reduction if self.o==0: u = ured else: if p==0: u = np.empty(self.n, dtype=np.float64) u[self.ki] = ured u[self.k] = y else: u = np.empty((self.n,p), dtype=np.float64) u[self.ki,:] = ured u[self.k,:] = y return u

min_quad_with_fixed(Q, c=None, A=None, b=None, k=None, y=None)

Solve the following quadratic program with linear constraints:

argmin_u  0.5 * tr(u.transpose()*Q*u) + tr(c.transpose()*u)
A*u == b
u[k] == y (if y is a 1-tensor) or u[k,:] == y) (if y is a 2-tensor)

Parameters:

Name Type Description Default
Q (n,n) symmetric sparse scipy csr_matrix

This matrix will be symmetrized if not exactly symmetric.

required
c None or scalar or (n,) numpy array or (n,p) numpy array

Assumed to be scalar 0 if None

None
A None or (m,n) sparse scipy csr_matrix

m=0 assumed if None

None
b None or scalar or (m,) numpy array or (m,p) numpy array

Assumed to be scalar 0 if None

None
k None or (o,) numpy array

o=0 assumed if None

None
y None or scalar or (o,) numpy array or (o,p) numpy array

Assumed to be scalar 0 if None

None

Returns:

Name Type Description
u (n,) numpy array or (n,p) numpy array

Solution to the optimization problem

Examples:

TODO