On the behavior of Lagrange multipliers in convex and nonconvex infeasible interior point methods

Abstract

We analyze sequences generated by interior point methods (IPMs) in convex and nonconvex settings. We prove that moving the primal feasibility at the same rate as the barrier parameter $\mu$ ensures the Lagrange multiplier sequence remains bounded, provided the limit point of the primal sequence has a Lagrange multiplier. This result does not require constraint qualifications. We also guarantee the IPM finds a solution satisfying strict complementarity if one exists. On the other hand, if the primal feasibility is reduced too slowly, then the algorithm converges to a point of minimal complementarity; if the primal feasibility is reduced too quickly and the set of Lagrange multipliers is unbounded, then the norm of the Lagrange multiplier tends to infinity. Our theory has important implications for the design of IPMs. Specifically, we show that IPOPT, an algorithm that does not carefully control primal feasibility has practical issues with the dual multipliers values growing to unnecessarily large values. Conversely, the one-phase IPM of Hinder and Ye (A one-phase interior point method for nonconvex optimization, 2018. arXiv:1801.03072), an algorithm that controls primal feasibility as our theory suggests, has no such issue.

Experimental details

The code for the experiments can be found at https://github.com/ohinder/Lagrange-multipliers-behavior.

1.1 Solvers

One-phase solver. For the well-behaved interior point solver, given a problem of the form

$$\begin{aligned} \text{ minimize }&f(x) \\ \text{ subject } \text{ to }&c(x) = 0 \\&x_{L} \le x \le x_{U}, \end{aligned}$$

we can re-write the constraints as

$$\begin{aligned} \text{ minimize }&f(x) \\ \text{ subject } \text{ to }&c(x) \le 0 \\ -&c(x) \le 0 \\&x_{L} \le x \le x_{U}. \end{aligned}$$

This gives a problem of the form

$$\begin{aligned} \text{ minimize }&f(x) \\ \text{ subject } \text{ to }&a(x) + s = 0 \\&s \ge 0, \end{aligned}$$

which we can pass to the one-phase solver.

The terms in Fig. 2 and Table 1 are given as follows:

• The infinity norm of the primal residual is given by $\Vert a(x) + s \Vert _{\infty }$.

• The infinity norm of the dual residual is measured by $\Vert \nabla {\mathcal {L}}(x,y) \Vert _{\infty }$.

• The infinity norm of complementarity is given by $\max _i s_i y_i$.

• We measure strict complementarity by $\min _i s_i + y_i$.

The optimality termination criterion of the one-phase IPM is

$$\begin{aligned} \max \left\{ \frac{100}{\max \{ \Vert y \Vert _{\infty }, 100 \}} \max \{ \Vert \nabla _{x} {\mathcal {L}}(x,y) \Vert _{\infty }, \Vert Sy \Vert _{\infty } \}, \Vert a(x) + s \Vert _{\infty } \right\} \le 10^{-6}. \end{aligned}$$

For more etails on the one-phase IPM see the paper [26] and code (https://github.com/ohinder/OnePhase.jl). The linear solver used was the default Julia Cholesky factorization (SuiteSparse).

Table 2 Solver options

Table 3 NETLIB problems where a solver failed

Table 4 Problems in NETLIB collection with a strict relative interior

IPOPT. We use IPOPT 3.12.4 with the linear solver MUMPS. Given any generic nonlinear problem, IPOPT rewrites it in the form (by adding slacks to inequalities, see [42])

$$\begin{aligned} \text{ minimize }&f(x) \\ \text{ subject } \text{ to }&c(x) = 0 \\&x_{L} \le x \le x_{U}. \end{aligned}$$

For practical reasons related to the interface we use [17], we do this reformulation ourselves. We then measure

• Primal feasibility by $\Vert c(x) \Vert _{\infty }$.

• Dual feasibility by $\Vert \nabla f(x) + \nabla c(x)^T \lambda - z_{L} + z_{U} \Vert _{\infty }$, where $z_{L}$ and $z_{U}$ are the dual multipliers corresponding to the constraint $x \ge l$ and $x \le u$ respectively (same notation as in [42]).

• Complementarity is given by $\max \{ \max _i((z_{L})_i (x_i - l_i)), \max _i( (z_{U})_i (x_i - u_i)) \}$.

• We measure strict complementarity by $\min \{ \min _i((z_{L})_i (x_i - l_i)), \min _i((z_{U})_i (x_i - l_i)) \}$.

The details of this computation can be found in the file ‘src/shared.jl’ in the function ‘add_solver_results!’.

The options chosen for the solvers are given in Table 2. We turn off the acceptable termination criterion for IPOPT to try to make the termination criterion of the algorithms as similar as possible.

1.2 NETLIB LP test details

The linear programs in the NETLIB linear programming collection come in the form $\min {c^T x}$ s.t. $Ax = b$, $l \le x \le u$. Table 3 shows which solver failed on which problem.

Table 4 shows when there is a feasible solution according to Gurobi when the bound constraints are tightened by $\delta$ i.e. find a solution to the system $Ax = b$ and $u - \delta \ge x \ge l + \delta$. We tried $\delta = 10^{-4}, 10^{-6}, 10^{-8}$ and obtained the same results with Gurobi’s feasibility tolerance set to $10^{-9}$. We found 29 problems with a feasible solution and 64 without a feasible solution in the NETLIB collection. We used Gurobi version 7.02.

1.3 Additional figures for nonconvex problems

This section gives plots of solver trajectories for the nonconvex problems of Sect. 4.2.

See Figs. 6, 7 and 8.

Fig. 6

Comparison on the problem of finding the intersection of two circles

Fig. 7

Comparison on a linear program with complementarity constraints

Fig. 8

Comparison on a toy drinking water network optimization problem