Nonlinear programming

About: Nonlinear programming is a research topic. Over the lifetime, 19486 publications have been published within this topic receiving 656602 citations. The topic is also known as: non-linear programming & NLP.

Perspective reformulations of mixed integer nonlinear programs with indicator variables

Abstract: We study mixed integer nonlinear programs (MINLP)s that are driven by a collection of indicator variables where each indicator variable controls a subset of the decision variables. An indicator variable, when it is “turned off”, forces some of the decision variables to assume fixed values, and, when it is “turned on”, forces them to belong to a convex set. Many practical MINLPs contain integer variables of this type. We first study a mixed integer set defined by a single separable quadratic constraint and a collection of variable upper and lower bound constraints, and a convex hull description of this set is derived. We then extend this result to produce an explicit characterization of the convex hull of the union of a point and a bounded convex set defined by analytic functions. Further, we show that for many classes of problems, the convex hull can be expressed via conic quadratic constraints, and thus relaxations can be solved via second-order cone programming. Our work is closely related with the earlier work of Ceria and Soares (Math Program 86:595–614, 1999) as well as recent work by Frangioni and Gentile (Math Program 106:225–236, 2006) and, Akturk et al. (Oper Res Lett 37:187–191, 2009). Finally, we apply our results to develop tight formulations of mixed integer nonlinear programs in which the nonlinear functions are separable and convex and in which indicator variables play an important role. In particular, we present computational results for three applications—quadratic facility location, network design with congestion, and portfolio optimization with buy-in thresholds—that show the power of the reformulation technique.

Adjustable Robust Real-Time Power Dispatch With Large-Scale Wind Power Integration

Abstract: In this paper, we propose a robust real-time wind power dispatch framework for coordinating wind farms, automatic generation control (AGC) units, and nonAGC units, which enables wind farms to operate flexibly using maximum power-point tracking strategies. Robust real-time dispatch is formulated as an adjustable robust optimization model incorporating an affinely adjustable controlling strategy compatible with AGC systems. The proposed model can be equivalently transformed to a nonlinear programming problem with linear constraints via duality. The proposed model can also be approximately reduced to a quadratic programming with the objective function simplified. Monte Carlo simulations are carried out to compare the performance of the proposed method against the conventional real-time dispatch scheme. The results show the proposed scheme is robust and reliable.

Ordinal optimisation approach for locating and sizing of distributed generation

Abstract: This study presents an ordinal optimisation (OO) method for specifying the locations and capacities of distributed generation (DG) such that a trade-off between loss minimisation and DG capacity maximisation is achieved. The OO approach consists of three main phases. First, the large search space of potential combinations of DG locations is represented by sampling a relatively small number of alternatives. Second, the objective function value for each of the sampled alternatives is evaluated using a crude but computationally efficient linear programming model. Third, the top-s alternatives from the crude model evaluation are simulated via an exact non-linear programming optimal power flow (OPF) programme to find the best DG locations and capacities. OO theory allows computing the size s of the selected subset such that it contains at least k designs from among the true top-g samples with a pre-specified alignment probability AP. This study discusses problem-specific approaches for sampling, crude model implementation and subset size selection. The approach is validated by comparing with recently published results of a hybrid genetic algorithm OPF applied to a 69-node distribution network operating under Ofgem (UK) financial incentives for distribution network operators.

Unconstrained Optimization of Real Functions in Complex Variables

Abstract: Nonlinear optimization problems in complex variables are frequently encountered in applied mathematics and engineering applications such as control theory, signal processing, and electrical engineering. Optimization of these problems often requires a first- or second-order approximation of the objective function to generate a new step or descent direction. However, such methods cannot be applied to real functions of complex variables because they are necessarily nonanalytic in their argument, i.e., the Taylor series expansion in their argument alone does not exist. To overcome this problem, the objective function is usually redefined as a function of the real and imaginary parts of its complex argument so that standard optimization methods can be applied. However, this approach may needlessly disguise any inherent structure present in the derivatives of such complex problems. Although little known, it is possible to construct an expansion of the objective function in its original complex variables by noti...