Numeric Tolerances¶

PICOS allows you to fine-tune how accurate your solution needs to be. Tolerances fall in three categories:

Feasibility tolerances, abbreviated fsb, control the magnitude of constraint violation that is tolerated. The integrality tolerance also falls into this category.
Optimality tolerances, abbreviated opt, control the maximum allowed deviation from the mathematically exact optimum solution and serve as a termination criterion. An exception is the the Simplex algorithm that uses the dual feasibility as its stopping criterion.
The remaining tolerances are used at intermediate steps of specific algorithms, such as the Markowitz threshold used in a pivoting strategy of the Simplex algoritm.

Solvers differ in how they measure deviations from the ideal values. Some bound absolute values while others consider the deviation in relation to the magnitude of the numbers that occur in the problem. PICOS abbreviates the former measurement with abs and the latter with rel. If both measurements are supported by a solver, then the standard approach is to allow values if they are sufficiently accurate according to either one.

If solvers use a single value for primal and dual feasibility but PICOS is configured to use differing accuracies, supplied in the options with the prim and dual abbreviations respectively, it will supply the smaller of both values to such solvers.

By default, PICOS overrides the solver’s default accuracies with common values, so that the choice of solver becomes transparent to you. Given that P is your problem instance, you can make PICOS respect the solvers’ individual choices as follows:

>>> import picos
>>> P = picos.Problem()
>>> P.options["*_tol"] = None

Comparison Table¶

The table shows what tolerance options are supported by PICOS and each solver, and what their respective default value is.

Option	PICOS	CPLEX	CVXOPT	ECOS	GLPK	Gurobi	MOSEK	SCIP	SMCP
abs_prim_fsb_tol	$10^{-8}$	$\text{SX:}~10^{-6}$	unused	unused ?	$\text{SX:}~10^{-7}~?$	$10^{-6}~?$	$\text{SX:}~10^{-6}$ $\text{LP:}~10^{-8}~?$ $\text{CP:}~10^{-8}~?$ $\text{QP:}~10^{-8}~?$ $\text{NL:}~10^{-8}~?$ $\text{IP:}~10^{-6}~?$	$\text{SX:}~10^{-6}$	unused
rel_prim_fsb_tol	$10^{-8}$	$\text{LQ:}~10^{-8}$ $\text{QC:}~10^{-8}$	$\text{CP:}~10^{-7}$ $\text{NL:}~10^{-7}$	$10^{-8}~?$	unused ?	unused ?		$10^{-6}$	$10^{-8}$
abs_dual_fsb_tol	$10^{-8}$	$\text{SX:}~10^{-6}$	unused	unused ?	$\text{SX:}~10^{-7}~?$	$10^{-6}$	$\text{SX:}~10^{-6}$ $\text{LP:}~10^{-8}~?$ $\text{CP:}~10^{-8}~?$ $\text{QP:}~10^{-8}~?$ $\text{NL:}~10^{-8}~?$	$\text{SX:}~10^{-7}$	unused
rel_dual_fsb_tol	$10^{-8}$	$\text{LQ:}~10^{-8}$ $\text{QC:}~10^{-8}$	$\text{CP:}~10^{-7}$ $\text{NL:}~10^{-7}$	$10^{-8}~?$	unused ?	unused	$\text{SX:}~10^{-12}$	$10^{-6}$	$10^{-8}$
abs_ipm_opt_tol	$10^{-8}$	unused	$\text{CP:}~10^{-7}$ $\text{NL:}~10^{-7}$	$10^{-8}$	unused	unused	unused	$0$	$10^{-6}$
rel_ipm_opt_tol	$10^{-8}$	$\text{LQ:}~10^{-8}$ $\text{QC:}~10^{-8}$	$\text{CP:}~10^{-6}$ $\text{NL:}~10^{-6}$	$10^{-8}$	unused	$\text{CO:}~10^{-8}$ $\text{QC:}~10^{-6}$	$\text{LP:}~10^{-8}$ $\text{CP:}~10^{-7}$ $\text{QP:}~10^{-7}$ $\text{NL:}~10^{-6}$	$0$	$10^{-6}$
abs_bnb_opt_tol	$10^{-6}$	$10^{-6}$	no IP	$10^{-6}$	unused	$10^{-10}$	$0$	$0$	no IP
rel_bnb_opt_tol	$10^{-4}$	$10^{-4}$	no IP	$10^{-3}$	$0$	$10^{-4}$	$10^{-4}$	$0$	no IP
integrality_tol	$10^{-5}$	$10^{-5}$	no IP	$10^{-4}$	$10^{-5}$	$10^{-5}$	$10^{-5}$	unused	no IP
markowitz_tol	`None`	$0.01$	no SX	no SX	$0.1$	$2^{-7}$	unused ?	unused	no SX

Pooled options

ECOS, CVXOPT, SCIP and SMCP merge rel_prim_fsb_tol and rel_dual_fsb_tol.
CPLEX merges rel_prim_fsb_tol, rel_dual_fsb_tol and rel_ipm_opt_tol.
SCIP appears to merge abs_ipm_opt_tol with abs_bnb_opt_tol and rel_ipm_opt_tol with rel_bnb_opt_tol with its limits/absgap and limits/gap options, respectively.

Legend

?	It is unclear whether an absolute or relative measure is used, or if an option is not available.
SX	Linear Programs via Simplex
LP	Linear Programs via Interior-Point Method
CP	Conic Programs
LQ	Linear and Quadratic Programs
QP	Quadratic Programs
QC	Quadratically Constrained (Quadratic) Programs
NL	Nonlinear Programs
IP	(Mixed) Integer Programs

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