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
60 CHAPTER 7. THE ANALYZERS 7.1.2 Objective The second part of the survey focuses on (the linear part of) the objective, summarizing the optimization sense and the coefficients’ absolute value range and distribution. The number of 0 (zero) coefficients is singled out (if any such variables are in the problem). The range is displayed using three terms: min |c|: The minimum absolute value among all coeffecients min |c|>0: The minimum absolute value among the nonzero coefficients max |c|: The maximum absolute value among the coefficients If some of these extrema turn out to be equal, the display is shortened accordingly: • If min |c| is greater than zero, the min |c|?0 term is obsolete and will not be displayed • If only one or two different coefficients occur this will be displayed using all and an explicit listing of the coefficients The absolute value distribution is displayed as a table summarizing the numbers by orders of magnitude (with a ratio of 10). Again, the number of variables with a coefficient of 0 (if any) is singled out. Each line of the table is headed by an interval (half-open intervals including their lower bounds), and is followed by the number of variables with their objective coefficient in this interval. Intervals with no elements are skipped. 7.1.3 Linear constraints The third part of the survey displays information on the nonzero coefficients of the linear constraint matrix. Following a brief summary of the matrix dimensions and the number of nonzero coefficients in total, three sections provide further details on how the nonzero coefficients are distributed by row-wise count (A i), by column-wise count (A|j), and by absolute value (|A(ij)|). Each section is headed by a brief display of the distribution’s range (min and max), and for the row/column-wise counts the corresponding densities are displayed too (in parentheses). The distribution tables single out three particularly interesting counts: zero, one, and two nonzeros per row/column; the remaining row/column nonzeros are displayed by orders of magnitude (ratio 2). For each interval the relative and accumulated relative counts are also displayed. Note that constraints may have both linear and quadratic terms, but the empty rows and columns reported in this part of the survey relate to the linear terms only. If empty rows and/or columns are found in the linear constraint matrix, the problem is analyzed further in order to determine if the
7.2. ANALYZING INFEASIBLE PROBLEMS 61 corresponding constraints have any quadratic terms or the corresponding variables are used in conic or quadratic constraints; cf. the last two examples of appendix 18. The distribution of the absolute values, |A(ij)|, is displayed just as for the objective coefficients described above. 7.1.4 Constraint and variable bounds The fourth part of the survey displays distributions for the absolute values of the finite lower and upper bounds for both constraints and variables. The number of bounds at 0 is singled out and, otherwise, displayed by orders of magnitude (with a ratio of 10). 7.1.5 Quadratic constraints The fifth part of the survey displays distributions for the nonzero elements in the gradient of the quadratic constraints, i.e. the nonzero row counts for the column vectors Qx . The table is similar to the tables for the linear constraints’ nonzero row and column counts described in the survey’s third part. Note: Quadratic constraints may also have a linear part, but that will be included in the linear constraints survey; this means that if a problem has one or more pure quadratic constraints, part three of the survey will report an equal number of linear constraint rows with 0 (zero) nonzeros, cf. the last example in appendix 18. Likewise, variables that appear in quadratic terms only will be reported as empty columns (0 nonzeros) in the linear constraint report. 7.1.6 Conic constraints The last part of the survey summarizes the model’s conic constraints. For each of the two types of cones, quadratic and rotated quadratic, the total number of cones are reported, and the distribution of the cones’ dimensions are displayed using intervals. Cone dimensions of 2, 3, and 4 are singled out. 7.2 Analyzing infeasible problems When developing and implementing a new optimization model, the first attempts will often be either infeasible, due to specification of inconsistent constraints, or unbounded, if important constraints have been left out. In this chapter we will • go over an example demonstrating how to locate infeasible constraints using the MOSEK infeasibility report tool, • discuss in more general terms which properties that may cause infeasibilities, and • present the more formal theory of infeasible and unbounded problems.
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7.2. ANALYZING INFEASIBLE PROBLEMS 61<br />
corresponding constraints have any quadratic terms or the corresponding variables are used in conic<br />
or quadratic constraints; cf. the last two examples of appendix 18.<br />
<strong>The</strong> distribution of the absolute values, |A(ij)|, is displayed just as for the objective coefficients<br />
described above.<br />
7.1.4 Constraint and variable bounds<br />
<strong>The</strong> fourth part of the survey displays distributions for the absolute values of the finite lower and upper<br />
bounds for both constraints and variables. <strong>The</strong> number of bounds at 0 is singled out and, otherwise,<br />
displayed by orders of magnitude (with a ratio of 10).<br />
7.1.5 Quadratic constraints<br />
<strong>The</strong> fifth part of the survey displays distributions for the nonzero elements in the gradient of the<br />
quadratic constraints, i.e. the nonzero row counts for the column vectors Qx . <strong>The</strong> table is similar to<br />
the tables for the <strong>line</strong>ar constraints’ nonzero row and column counts described in the survey’s third<br />
part.<br />
Note: Quadratic constraints may also have a <strong>line</strong>ar part, but that will be included in the <strong>line</strong>ar<br />
constraints survey; this means that if a problem has one or more pure quadratic constraints, part three<br />
of the survey will report an equal number of <strong>line</strong>ar constraint rows with 0 (zero) nonzeros, cf. the last<br />
example in appendix 18. Likewise, variables that appear in quadratic terms only will be reported as<br />
empty columns (0 nonzeros) in the <strong>line</strong>ar constraint report.<br />
7.1.6 Conic constraints<br />
<strong>The</strong> last part of the survey summarizes the model’s conic constraints. For each of the two types of<br />
cones, quadratic and rotated quadratic, the total number of cones are reported, and the distribution<br />
of the cones’ dimensions are displayed using intervals. Cone dimensions of 2, 3, and 4 are singled out.<br />
7.2 Analyzing infeasible problems<br />
When developing and implementing a new optimization model, the first attempts will often be either<br />
infeasible, due to specification of inconsistent constraints, or unbounded, if important constraints have<br />
been left out.<br />
In this chapter we will<br />
• go over an example demonstrating how to locate infeasible constraints using the <strong>MOSEK</strong> infeasibility<br />
report <strong>tool</strong>,<br />
• discuss in more general terms which properties that may cause infeasibilities, and<br />
• present the more formal theory of infeasible and unbounded problems.
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