The MOSEK Python optimizer API manual Version 7.0 (Revision 141)
Optimizer API for Python - Documentation - Mosek Optimizer API for Python - Documentation - Mosek
126 CHAPTER 10. PROBLEM FORMULATION AND SOLUTIONS are satisfied. (s c l ) ∗ i ((x c i) ∗ − li c ) = 0, i = 0, . . . , m − 1, (s c u) ∗ i (u c i − (x c i) ∗ ) = 0, i = 0, . . . , m − 1, (s x l ) ∗ j (x ∗ j − lj x ) = 0, j = 0, . . . , n − 1, (s x u) ∗ j (u x j − x ∗ j ) = 0, j = 0, . . . , n − 1, If (10.1) has an optimal solution and MOSEK solves the problem successfully, both the primal and dual solution are reported, including a status indicating the exact state of the solution. 10.1.2 Infeasibility for linear optimization 10.1.2.1 Primal infeasible problems If the problem (10.1) is infeasible (has no feasible solution), MOSEK will report a certificate of primal infeasibility: The dual solution reported is the certificate of infeasibility, and the primal solution is undefined. A certificate of primal infeasibility is a feasible solution to the modified dual problem maximize (l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u subject to A T y + s x l − s x u = 0, − y + s c l − s c u = 0, s c l , s c u, s x l , s x u ≥ 0, such that the objective value is strictly positive, i.e. a solution (10.4) to (10.4) so that (y ∗ , (s c l ) ∗ , (s c u) ∗ , (s x l ) ∗ , (s x u) ∗ ) (l c ) T (s c l ) ∗ − (u c ) T (s c u) ∗ + (l x ) T (s x l ) ∗ − (u x ) T (s x u) ∗ > 0. Such a solution implies that (10.4) is unbounded, and that its dual is infeasible. As the constraints to the dual of (10.4) is identical to the constraints of problem (10.1), we thus have that problem (10.1) is also infeasible. 10.1.2.2 Dual infeasible problems If the problem (10.2) is infeasible (has no feasible solution), MOSEK will report a certificate of dual infeasibility: The primal solution reported is the certificate of infeasibility, and the dual solution is undefined. A certificate of dual infeasibility is a feasible solution to the modified primal problem minimize subject to ˆlc ˆlx c T x ≤ Ax ≤ û c , ≤ x ≤ û x , (10.5)
10.2. CONIC QUADRATIC OPTIMIZATION 127 where { 0 if l c ˆlc i = i > −∞, − ∞ otherwise, { and û c 0 if u c i := i < ∞, ∞ otherwise, and { 0 if l x ˆlx j = j > −∞, − ∞ otherwise, { 0 if u and û x x j := j < ∞, ∞ otherwise, such that the objective value c T x is strictly negative. Such a solution implies that (10.5) is unbounded, and that its dual is infeasible. As the constraints to the dual of (10.5) is identical to the constraints of problem (10.2), we thus have that problem (10.2) is also infeasible. 10.1.2.3 Primal and dual infeasible case In case that both the primal problem (10.1) and the dual problem (10.2) are infeasible, MOSEK will report only one of the two possible certificates — which one is not defined (MOSEK returns the first certificate found). 10.2 Conic quadratic optimization Conic quadratic optimization is an extensions of linear optimization (see Section 10.1) allowing conic domains to be specified for subsets of the problem variables. A conic quadratic optimization problem can be written as minimize c T x + c f subject to l c ≤ Ax ≤ u c , l x ≤ x ≤ u x , x ∈ C, where set C is a Cartesian product of convex cones, namely C = C 1 × · · · ×C p . restriction, x ∈ C, is thus equivalent to (10.6) Having the domain x t ∈ C t ⊆ R nt , where x = (x 1 , . . . , x p ) is a partition of the problem variables. Please note that the n-dimensional Euclidean space R n is a cone itself, so simple linear variables are still allowed. MOSEK supports only a limited number of cones, specifically: • The R n set.
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126 CHAPTER 10. PROBLEM FORMULATION AND SOLUTIONS<br />
are satisfied.<br />
(s c l ) ∗ i ((x c i) ∗ − li c ) = 0, i = 0, . . . , m − 1,<br />
(s c u) ∗ i (u c i − (x c i) ∗ ) = 0, i = 0, . . . , m − 1,<br />
(s x l ) ∗ j (x ∗ j − lj x ) = 0, j = 0, . . . , n − 1,<br />
(s x u) ∗ j (u x j − x ∗ j ) = 0, j = 0, . . . , n − 1,<br />
If (10.1) has an optimal solution and <strong>MOSEK</strong> solves the problem successfully, both the primal and<br />
dual solution are reported, including a status indicating the exact state of the solution.<br />
10.1.2 Infeasibility for linear optimization<br />
10.1.2.1 Primal infeasible problems<br />
If the problem (10.1) is infeasible (has no feasible solution), <strong>MOSEK</strong> will report a certificate of primal<br />
infeasibility: <strong>The</strong> dual solution reported is the certificate of infeasibility, and the primal solution is<br />
undefined.<br />
A certificate of primal infeasibility is a feasible solution to the modified dual problem<br />
maximize (l c ) T s c l − (u c ) T s c u + (l x ) T s x l − (u x ) T s x u<br />
subject to A T y + s x l − s x u = 0,<br />
− y + s c l − s c u = 0,<br />
s c l , s c u, s x l , s x u ≥ 0,<br />
such that the objective value is strictly positive, i.e. a solution<br />
(10.4)<br />
to (10.4) so that<br />
(y ∗ , (s c l ) ∗ , (s c u) ∗ , (s x l ) ∗ , (s x u) ∗ )<br />
(l c ) T (s c l ) ∗ − (u c ) T (s c u) ∗ + (l x ) T (s x l ) ∗ − (u x ) T (s x u) ∗ > 0.<br />
Such a solution implies that (10.4) is unbounded, and that its dual is infeasible. As the constraints to<br />
the dual of (10.4) is identical to the constraints of problem (10.1), we thus have that problem (10.1) is<br />
also infeasible.<br />
10.1.2.2 Dual infeasible problems<br />
If the problem (10.2) is infeasible (has no feasible solution), <strong>MOSEK</strong> will report a certificate of dual<br />
infeasibility: <strong>The</strong> primal solution reported is the certificate of infeasibility, and the dual solution is<br />
undefined.<br />
A certificate of dual infeasibility is a feasible solution to the modified primal problem<br />
minimize<br />
subject to<br />
ˆlc<br />
ˆlx<br />
c T x<br />
≤ Ax ≤ û c ,<br />
≤ x ≤ û x ,<br />
(10.5)