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Th.is work presents a global convergence theory for a broad class of trust-reg.ion algorithms for the smooth nonlinear programm.ing problem with equality constra.ints. The ma.in result generalizes Powell's 1975 result for unconstra.ined trust-region algorithms. The trial step is characterized by very m.ild conditions on its normal and tangential components. The nonual component need not be computed accurately. The theory requires a quasi-normal c01nponent to satisfy a fraction of Cauchy decrease condition on the quadratic model of the linearized constra.ints. The tangential component then must satisfy a fraction of Cauchy decrease cond.ition on a quadratic model of the Lagrangian function in the translated tangent space of the constra.ints detenn.ined by the quasi-nonnal component. The La.grange multipliers estimates and the Hessian estimates are assumed only to be bounded. The other ma.in characteristic of th.is class of a.lgoritluns is that the step is evaluated by using the augmented La.gra.ng.ia.n as a merit function and the penalty para.meter is updated using the El-Alem scheme. The properties of the step together with the way that the penalty parameter is chosen are sufficient to establish global convergence. As an example, an algorithm is presented which can be viewed as a generalization of the Steiha.ug-Toint dogleg a.lgoritlun for the w1constra.ined case. It is based on a quadratic programming algorithm that uses a step in a quasi-nonna.l d.irection to the tangent space of the constra.ints and then does feasible conjugate reduced-gra.d.ient steps to solve the reduced quadratic program. This algorithm should cope quite well with large problems for which effective precond.itioners a.re known.

Th.is work presents a global convergence theory for a broad class of trust-reg.ion algorithms for the smooth nonlinear programm.ing problem with equality constra.ints. The ma.in result generalizes Powell's 1975 result for unconstra.ined trust-region algorithms. The trial step is characterized by very m.ild conditions on its normal and tangential components. The nonual component need not be computed accurately. The theory requires a quasi-normal c01nponent to satisfy a fraction of Cauchy decrease condition on the quadratic model of the linearized constra.ints. The tangential component then must satisfy a fraction of Cauchy decrease cond.ition on a quadratic model of the Lagrangian function in the translated tangent space of the constra.ints detenn.ined by the quasi-nonnal component. The La.grange multipliers estimates and the Hessian estimates are assumed only to be bounded. The other ma.in characteristic of th.is class of a.lgoritluns is that the step is evaluated by using the augmented La.gra.ng.ia.n as a merit function and the penalty para.meter is updated using the El-Alem scheme. The properties of the step together with the way that the penalty parameter is chosen are sufficient to establish global convergence. As an example, an algorithm is presented which can be viewed as a generalization of the Steiha.ug-Toint dogleg a.lgoritlun for the w1constra.ined case. It is based on a quadratic programming algorithm that uses a step in a quasi-nonna.l d.irection to the tangent space of the constra.ints and then does feasible conjugate reduced-gra.d.ient steps to solve the reduced quadratic program. This algorithm should cope quite well with large problems for which effective precond.itioners a.re known.

When formulated in mathematical terms, the problem of zoning a protected natural area subject to both box and spatial constraints results in a combinatorial optimization problem belonging to the NP‐hard class. This fact and the usual dimension of the problem (regularly in the tens of thousands order) suggest the need to apply a heuristic approach. In this contribution we describe a quantitative method for zoning protected natural areas based on a simulated annealing algorithm. Building upon previous work by Bos (1993), we introduce three main innovations (a quadratic function of distance between land units, a non‐symmetric matrix of compatibilities among uses, and a spatial connection constraint) that make the approach applicable for ecological purposes. When applied to solving small‐size simulated problems, the results were indistinguishable from those obtained via an exact, enumerative method. A coarse‐scale zoning of Talampaya National Park (Argentina) rendered maps remarkably similar to those produced by subject area experts using a non‐quantitative consensus‐seeking approach. Results are encouraging and show particular potential for the periodical update of zoning of protected natural areas. Such a capability is crucial for application in developing countries where both human and financial resources are usually scarce but still critical for updating zoning and management plans.

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