A nonlinear programming approach to a very large hydroelectric system optimization

A large scale hydroelectric system optimization is considered and solved by using a non-linear programming method. The largest numerical case involves approximately 6 000 variables, 4 000 linear equations, 11 000 linear and nonlinear inequality constraints and a nonlinear objective function. The solution method is based on

1. partial elimination of independent variables by solving linear equations,
2. essentially unconstrained optimization of a compound function that consists of the objective function, nonlinear inequality constraints and part of the linear inequality constraints. The compound function is obtained via penalty formulation.

The algorithm takes full advantage of the problem's structure and provides useful solutions for real life problems that, in general, are defined over empty feasible regions.     

A nonmonotone approximate sequence algorithm for unconstrained nonlinear optimization

A new nonmonotone algorithm is proposed and analyzed for unconstrained nonlinear optimization. The nonmonotone techniques applied in this algorithm are based on the estimate sequence proposed by Nesterov (Introductory Lectures on Convex Optimization: A Basic Course, 2004) for convex optimization. Under proper assumptions, global convergence of this algorithm is established for minimizing general nonlinear objective function with Lipschitz continuous derivatives. For convex objective function, this algorithm maintains the optimal convergence rate of convex optimization. In numerical experiments, this algorithm is specified by employing safe-guarded nonlinear conjugate gradient search directions. Numerical results show the nonmonotone algorithm performs significantly better than the corresponding monotone algorithm for solving the unconstrained optimization problems in the CUTEr (Bongartz et al. in ACM Trans. Math. Softw. 21:123–160, 1995) library.     

A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations

This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et?al. (SIAM J Sci Comput 32:3447?C3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhancements, such as functionality to avoid the computation of a normal step during every iteration. The implementation is included in the IPOPT software package paired with an iterative linear system solver and preconditioner provided in PARDISO. Numerical results on a large nonlinear optimization test set and two PDE-constrained optimization problems with control and state constraints are presented to illustrate that the implementation is robust and efficient for large-scale applications.     

On the complexity of finding first-order critical points in constrained nonlinear optimization

The complexity of finding $\epsilon $ -approximate first-order critical points for the general smooth constrained optimization problem is shown to be no worse that $O(\epsilon ^{-2})$ in terms of function and constraints evaluations. This result is obtained by analyzing the worst-case behaviour of a first-order short-step homotopy algorithm consisting of a feasibility phase followed by an optimization phase, and requires minimal assumptions on the objective function. Since a bound of the same order is known to be valid for the unconstrained case, this leads to the conclusion that the presence of possibly nonlinear/nonconvex inequality/equality constraints is irrelevant for this bound to apply.     

Some branch and bound techniques for nonlinear optimization

The range of nonlinear optimization problems which can be solved by Linear Programming and the Branch and Bound algorithm is extended by introducing Chains of Linked Ordered Sets and by allowing automatic interpolation of new variables. However this approach involves solving a succession of linear subproblems, whose solutions in general violate the logical requirements of the nonlinear formulation and may lie far from any local or global optimum. The paper describes techniques which are designed to improve the performance of the Branch and Bound algorithm on problems containing chains, and which also yield benefits in integer programming.Each linear subproblem is tightened towards the corresponding nonlinear problem by removing variables which must logically be nonbasic in any feasible solution. This is achieved by a presolve procedure, and also by post-optimal Lagrangian relaxation which tightens the bound on the objective function by assessing the cheapest way to satisfy any violated chain constraints. Frequently fewer subsequent branches are required to find a feasible solution or to prove infeasibility.Formerly of Scicon Ltd.     

A new inexact SQP algorithm for nonlinear systems of mixed equalities and inequalities

Traditional inexact SQP algorithm can only solve equality constrained optimization (Byrd et al. Math. Program. 122, 273–299 2010). In this paper, we propose a new inexact SQP algorithm with affine scaling technique for nonlinear systems of mixed equalities and inequalities, which arise in complementarity and variational inequalities. The nonlinear systems are transformed into a special nonlinear optimization with equality and bound constraints, and then we give a new inexact SQP algorithm for solving it. The new algorithm equipped with affine scaling technique does not require a quadratic programming subproblem with inequality constraints. The search direction is computed by solving one linear system approximately using iterative linear algebra techniques. Under mild assumptions, we discuss the global convergence. The preliminary numerical results show the effectiveness of the proposed algorithm.     

Design and analysis of two discrete-time ZD algorithms for time-varying nonlinear minimization

Nonlinear minimization, as a subcase of nonlinear optimization, is an important issue in the research of various intelligent systems. Recently, Zhang et al. developed the continuous-time and discrete-time forms of Zhang dynamics (ZD) for time-varying nonlinear minimization. Based on this previous work, another two discrete-time ZD (DTZD) algorithms are proposed and investigated in this paper. Specifically, the resultant DTZD algorithms are developed for time-varying nonlinear minimization by utilizing two different types of Taylor-type difference rules. Theoretically, each steady-state residual error in the DTZD algorithm changes in an O(τ ³) manner with τ being the sampling gap. Comparative numerical results are presented to further substantiate the efficacy and superiority of the proposed DTZD algorithms for time-varying nonlinear minimization.     

A new Filter-Levenberg-Marquardt method with disturbance for solving nonlinear complementarity problems

Recently, filter methods are extensively studied to handle nonlinear programming problems. Because of good numerical results, filter techniques are attached importance to. The nonlinear complementarity problem can be reformulated as the least l₂-norm solution of an optimization problem. In this paper, basing on the filter technique and the new smoothing function, we present a new Filter-Levenberg-Marquardt method for solving the equation system with the disturbance . Under the assumption that the lever set of the problem is compact, we prove its global convergence.     

G-Perfect nonlinear functions

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both -valued and -valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.      

Error bounds for mixed integer nonlinear optimization problems

We introduce a-posteriori and a-priori error bounds for optimality and feasibility of a point generated as the rounding of an optimal point of the NLP relaxation of a mixed-integer nonlinear optimization problem. Our analysis mainly bases on the construction of a tractable approximation of the so-called grid relaxation retract. Under appropriate Lipschitz assumptions on the defining functions, we thereby generalize and slightly improve results for the mixed-integer linear case from Stein (Mathematical Programming, 2015, doi: 10.1007/s10107-015-0872-7). In particular, we identify cases in which the optimality and feasibility errors tend to zero at an at least linear rate for increasingly refined meshes.     

Continuous and discrete Zhang dynamics for real-time varying nonlinear optimization

Online solution of time-varying nonlinear optimization problems is considered an important issue in the fields of scientific and engineering research. In this study, the continuous-time derivative (CTD) model and two gradient dynamics (GD) models are developed for real-time varying nonlinear optimization (RTVNO). A continuous-time Zhang dynamics (CTZD) model is then generalized and investigated for RTVNO to remedy the weaknesses of CTD and GD models. For possible digital hardware realization, a discrete-time Zhang dynamics (DTZD) model, which can be further reduced to Newton-Raphson iteration (NRI), is also proposed and developed. Theoretical analyses indicate that the residual error of the CTZD model has an exponential convergence, and that the maximum steady-state residual error (MSSRE) of the DTZD model has an O(τ²) pattern with τ denoting the sampling gap. Simulation and numerical results further illustrate the efficacy and advantages of the proposed CTZD and DTZD models for RTVNO.     

Oscillations of nonlinear partial difference systems

This paper studies the following two-dimensional nonlinear partial difference systems

     

The nonlinear optimization problem in heat conduction

In this paper we study the existence and geometric properties of an optimal configuration to a nonlinear optimization problem in heat conduction. The quantity to be minimized is , where D is a fixed domain. A nonconstant temperature distribution is prescribed on and a volume constraint on the set where the temperature is positive is imposed. Among other regularity properties of an optimal configuration, we prove analyticity of the free boundary.Received: 6 October 2004, Accepted: 19 October 2004, Published online: 22 December 2004     

Backward nonlinear expectation equations

Building on an abstract framework for dynamic nonlinear expectations that comprises g-, G- and random G-expectations, we develop a theory of backward nonlinear expectation equations of the form

$$\begin{aligned} X_t = {\mathcal {E}}_t \Bigl [{\textstyle \int _t^T} g(s,X) \mu ({\mathrm {d}}s) + \xi \Bigr ], \quad t \in [0,T]. \end{aligned}$$

We provide existence, uniqueness, and stability results and establish convergence of the associated discrete-time nonlinear aggregations. As an application, we construct continuous-time recursive utilities under ambiguity and identify the corresponding utility processes as limits of discrete-time recursive utilities.

     

An optimization problem for nonlinear Steklov eigenvalues with a boundary potential

In this paper, we analyze an optimization problem for the first (nonlinear) Steklov eigenvalue plus a boundary potential with respect to the potential function which is assumed to be uniformly bounded and with fixed L¹$L^1$-norm.     

Global saddle points of nonlinear augmented Lagrangian functions

We notice that the results for the existence of global (local) saddle points of augmented Lagrangian functions in the literature were only sufficient conditions of some special types of augmented Lagrangian. In this paper, we introduce a general class of nonlinear augmented Lagrangian functions for constrained optimization problem. In two different cases, we present sufficient and necessary conditions for the existence of global saddle points. Moreover, as corollaries of the two results above, we not only obtain sufficient and necessary conditions for the existence of global saddle points of some special types of augmented Lagrangian functions mentioned in the literature, but also give some weaker sufficient conditions than the ones in the literature. Compared with our recent work (Wang et al. in Math Oper Res 38:740–760, 2013), the nonlinear augmented Lagrangian functions in this paper are more general and the results in this paper are original. We show that some examples (such as improved barrier augmented Lagrangian) satisfy the assumptions of this paper, but not available in Wang et al. (2013).