MIPS – Matpower Interior Point Solver

Beginning with version 4, Matpower includes a new primal-dual interior point solver called MIPS, for Matpower Interior Point Solver. It is implemented in pure-Matlab code, derived from the MEX implementation of the algorithms described in [8, 43].

This solver has application outside of Matpower to general nonlinear optimization problems of the following form:

where the input and output arguments are described in Tables A-1 and A-2, respectively. Alternatively, the input arguments can be packaged as ﬁelds in a problem struct and passed in as a single argument, where all ﬁelds except f_fcn and x0 are optional.

Table A-2:Output Arguments for mips

name

description

solution vector

ﬁnal objective function value

exitflag

exit ﬂag

1 –	ﬁrst order optimality conditions satisﬁed
0 –	maximum number of iterations reached
-1 –	numerically failed
–

output

output struct with ﬁelds

iterations	number of iterations performed
hist	struct array with trajectories of the following: feascond, gradcond, compcond, costcond, gamma, stepsize, obj, alphap, alphad
message	exit message

lambda

struct containing the Langrange and Kuhn-Tucker multipliers on the constraints, with ﬁelds:

eqnonlin	nonlinear equality constraints
ineqnonlin	nonlinear inequality constraints
mu_l	lower (left-hand) limit on linear constraints
mu_u	upper (right-hand) limit on linear constraints
lower	lower bound on optimization variables
upper	upper bound on optimization variables

The calling syntax is nearly identical to that used by fmincon from Matlab’s Optimization Toolbox. The primary diﬀerence is that the linear constraints are speciﬁed in terms of a single doubly-bounded linear function ( $l ≤ Ax ≤ u$ ) as opposed to separate equality constrained ( $Aeqx = beq$ ) and upper bounded ( $Ax ≤ b$ ) functions. Internally, equality constraints are handled explicitly and determined at run-time based on the values of $l$ and $u$ .

The user-deﬁned functions for evaluating the objective function, constraints and Hessian are identical to those required by fmincon, with one exception described below for the Hessian evaluation function. Speciﬁcally, f_fcn should return f as the scalar objective function value $f(x)$ , df as an $n × 1$ vector equal to $∇f$ and, unless gh_fcn is provided and the Hessian is computed by hess_fcn, d2f as an $n × n$ matrix equal to the Hessian $∂2f ∂x2$ . Similarly, the constraint evaluation function gh_fcn must return the $m × 1$ vector of nonlinear equality constraint violations $g(x)$ , the $p × 1$ vector of nonlinear inequality constraint violations $h(x)$ along with their gradients in dg and dh. Here dg is an $n × m$ matrix whose $jth$ column is $∇gj$ and dh is $n × p$ , with $th j$ column equal to $∇hj$ . Finally, for cases with nonlinear constraints, hess_fcn returns the $n × n$ Hessian $2 ∂∂xℒ2$ of the Lagrangian function

for given values of the multipliers $λ$ and $μ$ , where $σ$ is the cost_mult scale factor for the objective function. Unlike fmincon, mips passes this scale factor to the Hessian evaluation function in the 3^rd argument.

The use of nargout in f_fcn and gh_fcn is recommended so that the gradients and Hessian are only computed when required.

name	description
f_fcn	Handle to a function that evaluates the objective function, its gradients and Hessian^‡ for a given value of $x$ . Calling syntax for this function:
	[f, df, d2f] = f_fcn(x)
x0	Starting value of optimization vector $x$ .
A, l, u	Deﬁne the optional linear constraints $l ≤ Ax ≤ u$ . Default values for the elements of l and u are -Inf and Inf, respectively.
xmin, xmax	Optional lower and upper bounds on the $x$ variables, defaults are -Inf and Inf, respectively.
gh_fcn	Handle to function that evaluates the optional nonlinear constraints and their gradients for a given value of $x$ . Calling syntax for this function is:
	[h, g, dh, dg] = gh_fcn(x)
hess_fcn	Handle to function that computes the Hessian^‡ of the Lagrangian for given values of $x$ , $λ$ and $μ$ , where $λ$ and $μ$ are the multipliers on the equality and inequality constraints, $g$ and $h$ , respectively. The calling syntax for this function is:
	Lxx = hess_fcn(x, lam, cost_mult),
	where $λ$ = lam.eqnonlin, $μ$ = lam.ineqnonlin and cost_mult is a parameter used to scale the objective function
opt	Optional options structure with ﬁelds, all of which are also optional, described in Table A-3.
problem	Alternative, single argument input struct with ﬁelds corresponding to arguments above.

A MIPS – Matpower Interior Point Solver