We consider the standard optimistic bilevel optimization problem, in particular upper- and lower-level constraints can be coupled. By means of the lower-level value function, the problem is transformed into a single-level optimization problem with a penalization of the value function constraint. For treating the latter problem, we develop a framework that does not rely on the direct computation of the lower-level value function or its derivatives. For each penalty parameter, the framework leads to a semismooth system of equations. This allows us to extend the semismooth Newton method to bilevel optimization. Besides global convergence properties of the method, we focus on achieving local superlinear convergence to a solution of the semismooth system. To this end, we formulate an appropriate CD-regularity assumption and derive sufficient conditions so that it is fulfilled. Moreover, we develop conditions to guarantee that a solution of the semismooth system is a local solution of the bilevel optimization problem. Extensive numerical experiments on 124 examples of nonlinear bilevel optimization problems from the literature show that this approach exhibits a remarkable performance, where only a few penalty parameters need to be considered.

Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments

Recently, a local framework of Newton-type methods for constrained systems of equations has been developed. Applied to the solution of Karush–Kuhn–Tucker (KKT) systems, the framework enables local ...

A special complementarity function revisited

This paper concerns the issue of asymptotic acceptance of the true Hessian and the full step by the sequential quadratic programming algorithm for equality-constrained optimization problems. In order to enforce global convergence, the algorithm is equipped with a standard Armijo linesearch procedure for a nonsmooth exact penalty function. The specificity of considerations here is that the standard assumptions for local superlinear convergence of the method may be violated. The analysis focuses on the case when there exist critical Lagrange multipliers, and does not require regularity assumptions on the constraints or satisfaction of second-order sufficient optimality conditions. The results provide a basis for application of known acceleration techniques, such as extrapolation, and allow the formulation of algorithms that can outperform the standard SQP with BFGS approximations of the Hessian on problems with degenerate constraints. This claim is confirmed by some numerical experiments.

Accelerating convergence of a globalized sequential quadratic programming method to critical Lagrange multipliers

In this paper, we propose a new method that combines the inexact Newton method with a procedure to obtain a feasible inexact projection for solving constrained smooth and nonsmooth equations. The local convergence theorems are established under the assumption of smoothness or semismoothness of the function that defines the equation and its regularity at the solution. In particular, we show that a sequence generated by the method converges to a solution with linear, superlinear, or quadratic rate, under suitable conditions. Moreover, some numerical experiments are reported to illustrate the practical behavior of the proposed method.

Inexact Newton method with feasible inexact projections for solving constrained smooth and nonsmooth equations

Introduction. Recently, much attention has been paid to the issues of long-term development of specialized fire and rescue units of the Federal Fire Service of the State Fire Service. In this regard, there is a need to develop criteria to justify the use of a particular service as part of specialized fire and rescue units. Therefore, the objective of this study is to develop a mathematical model to justify the need to use radiation and chemical protection services as part of specialized fire and rescue units in the subjects of the Russian Federation.Materials and Methods. Justification of the need to use radiation and chemical protection services as part of specialized fire and rescue units has been carried out using the theory of fuzzy sets. The mathematical model takes into account the climatic and geographical features of the subjects, indicators of social, technical and economic development, and the risks of emergencies and fires. It also takes into account the availability of forces and means of a Unified State system for the prevention and liquidation of emergency situations in each subject of the Russian Federation. In total, 15 indicators were selected that characterize the need to use radiation and chemical protection services as part of specialized fire and rescue units. A desirability function is defined for each indicator, which shows which values of the indicator are the most acceptable from the point of view of the need to use radiation and chemical protection services as part of specialized fire and rescue units.Results. Using the developed model, the subjects of the Russian Federation are identified in which the need for radiation and chemical protection service as part of specialized fire and rescue units is the highest. It is proposed to create a radiation and chemical protection service of the 1st category in the Moscow, Sverdlovsk and Rostov regions, in the Krasnoyarsk and Primorsky Territories and in St. Petersburg. In 21 subjects it is proposed to use the radiation and chemical protection service of the 2nd category. In other subjects, it is proposed to assign the 3rd category to the radiation and chemical protection service.Discussion and Conclusion. The mathematical model developed using the theory of fuzzy sets will allow a more differentiated approach to the creation of a radiation and chemical protection service as part of specialized fire and rescue units and increase the efficiency of the functioning of this service and specialized fire and rescue units as a whole. The presented model can be applied to justify the need to use other services and groups as part of specialized fire and rescue units.

Justification of the Need to Use the Radiation and Chemical Protection Service as Part of Special Fire and Rescue Units in the Subjects of the Russian Federation

The LP-Newton method for constrained equations, introduced some years ago, has powerful properties of local superlinear convergence, covering both possibly nonisolated solutions and possibly nonsmooth equation mappings. A related globally convergent algorithm, based on the LP-Newton subproblems and linesearch for the equation’s infinity norm residual, has recently been developed. In the case of smooth equations, global convergence of this algorithm to B-stationary points of the residual over the constraint set has been shown, which is a natural result: nothing better should generally be expected in variational settings. However, for the piecewise smooth case only a property weaker than B-stationarity could be guaranteed. In this paper, we develop a procedure for piecewise smooth equations that avoids undesirable accumulation points, thus achieving the intended property of B-stationarity.

A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points

It is a well-known phenomenon that the presence of critical Lagrange multipliers in constrained optimization problems may cause a deterioration of the convergence speed of primal-dual Newton-type m...

Wladimir Scheck

Papers

A globally convergent LP-Newton method for piecewise smooth constrained equations: escaping nonstationary accumulation points

Adjusting Dual Iterates in the Presence of Critical Lagrange Multipliers