The MOSEK command line tool Version 7.0 (Revision 141)
The MOSEK command line tool. Version 7.0 ... - Documentation The MOSEK command line tool. Version 7.0 ... - Documentation
68 CHAPTER 7. THE ANALYZERS 7.2.5 Theory concerning infeasible problems This section discusses the theory of infeasibility certificates and how MOSEK uses a certificate to produce an infeasibility report. In general, MOSEK solves the problem where the corresponding dual problem is (7.3) minimize c T x + c f subject to l c ≤ Ax ≤ u c , l x ≤ x ≤ u x 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 + c f subject to A T y + s x l − s x u = c, − y + s c l − s c u = 0, s c l , s c u, s x l , s x u ≥ 0. We use the convension that for any bound that is not finite, the corresponding dual variable is fixed at zero (and thus will have no influence on the dual problem). For example (7.4) l x j = −∞ ⇒ (s x l ) j = 0 7.2.6 The certificate of primal infeasibility A certificate of primal infeasibility is any solution to the homogenized dual problem with a positive objective value. That is, (s c∗ l , s c∗ 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. u , s x∗ l , s x∗ u ) is a certificate of primal infeasibility if and (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 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. The well-known Farkas Lemma tells us that (7.3) is infeasible if and only if a certificate of primal infeasibility exists. Let (s c∗ l , s c∗ u , s x∗ l , s x∗ u ) be a certificate of primal infeasibility then
7.2. ANALYZING INFEASIBLE PROBLEMS 69 (s c∗ l ) i > 0((s c∗ u ) i > 0) implies that the lower (upper) bound on the i th constraint is important for the infeasibility. Furthermore, (s x∗ l ) j > 0((s x∗ u ) i > 0) implies that the lower (upper) bound on the j th variable is important for the infeasibility. 7.2.7 The certificate of dual infeasibility A certificate of dual infeasibility is any solution to the problem minimize c T x subject to ¯lc ≤ Ax ≤ ū c , ¯lx ≤ x ≤ ū x with negative objective value, where we use the definitions and { 0, l c ¯lc i := i > −∞, − ∞, otherwise, , ū c i := { 0, u c i < ∞, ∞, otherwise, { 0, l x ¯lx i := i > −∞, − ∞, otherwise, and ū x i := { 0, u x i < ∞, ∞, otherwise. Stated differently, a certificate of dual infeasibility is any x ∗ such that ¯lc ¯lx c T x ∗ < 0, ≤ Ax ∗ ≤ ū c , (7.5) ≤ x ∗ ≤ ū x The well-known Farkas Lemma tells us that (7.4) is infeasible if and only if a certificate of dual infeasibility exists. Note that if x ∗ is a certificate of dual infeasibility then for any j such that variable j is involved in the dual infeasibility. x ∗ j ≠ 0,
- Page 33 and 34: 3.10. SENSITIVITY ANALYSIS 17 • .
- Page 35 and 36: Chapter 4 Problem formulation and s
- Page 37 and 38: 4.1. LINEAR OPTIMIZATION 21 be a pr
- Page 39 and 40: 4.2. CONIC QUADRATIC OPTIMIZATION 2
- Page 41 and 42: 4.2. CONIC QUADRATIC OPTIMIZATION 2
- Page 43 and 44: 4.3. SEMIDEFINITE OPTIMIZATION 27 4
- Page 45 and 46: 4.5. GENERAL CONVEX OPTIMIZATION 29
- Page 47 and 48: 4.5. GENERAL CONVEX OPTIMIZATION 31
- Page 49 and 50: Chapter 5 The optimizers for contin
- Page 51 and 52: 5.1. HOW AN OPTIMIZER WORKS 35 5.1.
- Page 53 and 54: 5.2. LINEAR OPTIMIZATION 37 5.2.2 T
- Page 55 and 56: 5.2. LINEAR OPTIMIZATION 39 Wheneve
- Page 57 and 58: 5.2. LINEAR OPTIMIZATION 41 5.2.2.3
- Page 59 and 60: 5.2. LINEAR OPTIMIZATION 43 • Rai
- Page 61 and 62: 5.5. NONLINEAR CONVEX OPTIMIZATION
- Page 63 and 64: 5.6. SOLVING PROBLEMS IN PARALLEL 4
- Page 65 and 66: Chapter 6 The optimizers for mixed-
- Page 67 and 68: 6.3. THE MIXED-INTEGER CONIC OPTIMI
- Page 69 and 70: 6.5. TERMINATION CRITERION 53 The f
- Page 71 and 72: 6.7. UNDERSTANDING SOLUTION QUALITY
- Page 73 and 74: Chapter 7 The analyzers 7.1 The pro
- Page 75 and 76: 7.1. THE PROBLEM ANALYZER 59 Constr
- Page 77 and 78: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 79 and 80: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 81 and 82: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 83: 7.2. ANALYZING INFEASIBLE PROBLEMS
- Page 87 and 88: Chapter 8 Sensitivity analysis 8.1
- Page 89 and 90: 8.4. SENSITIVITY ANALYSIS FOR LINEA
- Page 91 and 92: 8.5. SENSITIVITY ANALYSIS WITH THE
- Page 93 and 94: 8.5. SENSITIVITY ANALYSIS WITH THE
- Page 95 and 96: Chapter 9 Parameters Parameters gro
- Page 97 and 98: 81 • MSK SPAR ITR SOL FILE NAME.
- Page 99 and 100: 83 • MSK DPAR INTPNT NL TOL REL G
- Page 101 and 102: 85 • MSK IPAR LOG MIO. Controls t
- Page 103 and 104: 87 • MSK DPAR INTPNT NL MERIT BAL
- Page 105 and 106: 89 • MSK IPAR INFEAS PREFER PRIMA
- Page 107 and 108: 91 • MSK IPAR SIM SAVE LU. Contro
- Page 109 and 110: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 111 and 112: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 113 and 114: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 115 and 116: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 117 and 118: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 119 and 120: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 121 and 122: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 123 and 124: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 125 and 126: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 127 and 128: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 129 and 130: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 131 and 132: 9.1. MSKDPARAME: DOUBLE PARAMETERS
- Page 133 and 134: 9.2. MSKIPARAME: INTEGER PARAMETERS
68 CHAPTER 7. THE ANALYZERS<br />
7.2.5 <strong>The</strong>ory concerning infeasible problems<br />
This section discusses the theory of infeasibility certificates and how <strong>MOSEK</strong> uses a certificate to<br />
produce an infeasibility report. In general, <strong>MOSEK</strong> solves the problem<br />
where the corresponding dual problem is<br />
(7.3)<br />
minimize<br />
c T x + c f<br />
subject to l c ≤ Ax ≤ u c ,<br />
l x ≤ x ≤ u x<br />
maximize (l c ) T s c l − (u c ) T s c u<br />
+ (l x ) T s x l − (u x ) T s x u + c f<br />
subject to A T y + s x l − s x u = c,<br />
− y + s c l − s c u = 0,<br />
s c l , s c u, s x l , s x u ≥ 0.<br />
We use the convension that for any bound that is not finite, the corresponding dual variable is fixed<br />
at zero (and thus will have no influence on the dual problem). For example<br />
(7.4)<br />
l x j = −∞ ⇒ (s x l ) j = 0<br />
7.2.6 <strong>The</strong> certificate of primal infeasibility<br />
A certificate of primal infeasibility is any solution to the homogenized dual problem<br />
with a positive objective value. That is, (s c∗<br />
l<br />
, s c∗<br />
maximize (l c ) T s c l − (u c ) T s c u<br />
+ (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 />
u , s x∗<br />
l<br />
, s x∗<br />
u ) is a certificate of primal infeasibility if<br />
and<br />
(l c ) T s c∗<br />
l<br />
− (u c ) T s c∗<br />
u<br />
+ (l x ) T s x∗<br />
l<br />
− (u x ) T s x∗<br />
u > 0<br />
A T y + s x∗<br />
l − s x∗<br />
u = 0,<br />
− y + s c∗<br />
l − s c∗<br />
u = 0,<br />
s c∗<br />
l , s c∗<br />
u , s x∗<br />
l , s x∗<br />
u ≥ 0.<br />
<strong>The</strong> well-known Farkas Lemma tells us that (7.3) is infeasible if and only if a certificate of primal<br />
infeasibility exists.<br />
Let (s c∗<br />
l<br />
, s c∗<br />
u , s x∗<br />
l<br />
, s x∗<br />
u ) be a certificate of primal infeasibility then