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The supporting hyperplane optimization toolkit for convex MINLP
In this paper, an open-source solver for mixed-integer nonlinear programming (MINLP) problems is presented. The Supporting Hyperplane Optimization Toolkit (SHOT) combines a dual strategy based on polyhedral outer approximations (POA) with primal heuristics. The POA is achieved by expressing the nonlinear feasible set of the MINLP problem with linearizations obtained with the extended supporting hyperplane (ESH) and extended cutting plane (ECP) algorithms. The dual strategy can be tightly integrated with the mixed-integer programming (MIP) subsolver in a so-called single-tree manner, i.e., only a single MIP optimization problem is solved, where the polyhedral linearizations are added as lazy constraints through callbacks in the MIP solver. This enables the MIP solver to reuse the branching tree in each iteration, in contrast to most other POA-based methods. SHOT is available as a COIN-OR open-source project, and it utilizes a flexible task-based structure making it easy to extend and modify. It is currently available in GAMS, and can be utilized in AMPL, Pyomo and JuMP as well through its ASL interface. The main functionality and solution strategies implemented in SHOT are described in this paper, and their impact on the performance are illustrated through numerical benchmarks on 406 convex MINLP problems from the MINLPLib problem library. Many of the features introduced in SHOT can be utilized in other POA-based solvers as well. To show the overall effectiveness of SHOT, it is also compared to other state-of-the-art solvers on the same benchmark set.
Journal Article
Robust Optimization
Robust optimization is still a relatively new approach to optimization problems affected by uncertainty, but it has already proved so useful in real applications that it is difficult to tackle such problems today without considering this powerful methodology. Written by the principal developers of robust optimization, and describing the main achievements of a decade of research, this is the first book to provide a comprehensive and up-to-date account of the subject.
Robust optimization is designed to meet some major challenges associated with uncertainty-affected optimization problems: to operate under lack of full information on the nature of uncertainty; to model the problem in a form that can be solved efficiently; and to provide guarantees about the performance of the solution.
The book starts with a relatively simple treatment of uncertain linear programming, proceeding with a deep analysis of the interconnections between the construction of appropriate uncertainty sets and the classical chance constraints (probabilistic) approach. It then develops the robust optimization theory for uncertain conic quadratic and semidefinite optimization problems and dynamic (multistage) problems. The theory is supported by numerous examples and computational illustrations.
An essential book for anyone working on optimization and decision making under uncertainty,Robust Optimizationalso makes an ideal graduate textbook on the subject.
eBook
Research on real-time scheduling and collaborative control of collaborative planning of power supply and new energy based on nonlinear programming algorithm in power system
With the rapid expansion of the installed scale of new energy(NE), it poses great pressure on the safe and stable operation of power grid (PG), and it is urgent to carry out research on the coordinated development of NE power supply and PG. This paper takes the coordinated development of power generation, transmission and distribution and dispatching as the research object, and adopts the multi-source coordinated development evaluation model of nonlinear programming algorithm in power system to carry out the coordinated evaluation research of the coordinated development of NE sources and regional networks. In the case of balanced access to electricity, wind power is 6.233%, solar photovoltaic power generation is 2.12%, hydropower is 0.835%, and thermal power is 90.812%. This paper is helpful for real-time scheduling of power supply and NE collaborative planning.
Journal Article
Alternative regularizations for Outer-Approximation algorithms for convex MINLP
In this work, we extend the regularization framework from Kronqvist et al. (Math Program 180(1):285–310, 2020) by incorporating several new regularization functions and develop a regularized single-tree search method for solving convex mixed-integer nonlinear programming (MINLP) problems. We propose a set of regularization functions based on distance metrics and Lagrangean approximations, used in the projection problem for finding new integer combinations to be used within the Outer-Approximation (OA) method. The new approach, called Regularized Outer-Approximation (ROA), has been implemented as part of the open-source Mixed-integer nonlinear decomposition toolbox for Pyomo—MindtPy. We compare the OA method with seven regularization function alternatives for ROA. Moreover, we extend the LP/NLP Branch and Bound method proposed by Quesada and Grossmann (Comput Chem Eng 16(10–11):937–947, 1992) to include regularization in an algorithm denoted RLP/NLP. We provide convergence guarantees for both ROA and RLP/NLP. Finally, we perform an extensive computational experiment considering all convex MINLP problems in the benchmark library MINLPLib. The computational results show clear advantages of using regularization combined with the OA method.
Journal Article
Scatter Search and Local NLP Solvers: A Multistart Framework for Global Optimization
The algorithm described here, called OptQuest/NLP or OQNLP, is a heuristic designed to find global optima for pure and mixed integer nonlinear problems with many constraints and variables, where all problem functions are differentiable with respect to the continuous variables. It uses OptQuest, a commercial implementation of scatter search developed by OptTek Systems, Inc., to provide starting points for any gradient-based local solver for nonlinear programming (NLP) problems. This solver seeks a local solution from a subset of these points, holding discrete variables fixed. The procedure is motivated by our desire to combine the superior accuracy and feasibility-seeking behavior of gradient-based local NLP solvers with the global optimization abilities of OptQuest. Computational results include 155 smooth NLP and mixed integer nonlinear program (MINLP) problems due to Floudas et al. (1999), most with both linear and nonlinear constraints, coded in the GAMS modeling language. Some are quite large for global optimization, with over 100 variables and 100 constraints. Global solutions to almost all problems are found in a small number of local solver calls, often one or two.
Journal Article
Reliability optimization and redundancy allocation for fire extinguisher drone using hybrid PSO–GWO
Reliability–redundancy allocation problem (RRAP) plays a vital role in reliability improvement and designing of systems which depend on the arrangement of components, reliability of the components, and redundancy allocation for the components. Higher reliability is the primary requisite for essential systems such as fire extinguisher drones (FEDs) which are very valuable for firefighters in tackling emergencies in non-reachable areas. In this work, a FED is considered with the aim of system designing for maximum reliability while considering the limited availability of resources such as volume, cost, and weight of the system. A total of five possible arrangements of the redundant components are investigated, and a mixed-integer nonlinear programming problem is solved for system reliability optimization. For optimization purposes, a recently developed metaheuristic hybrid particle swarm grey wolf optimizer (HPSGWO) is implemented. The HPSGWO is a powerful fusion of PSO’s exploitation property and GWO’s exploration property. Solving RRAP by using HPSGWO provides 99% reliability of the proposed FED under the limited availability of resources. To validate the superiority of the HPSGWO, a comparative study is explained.
Journal Article
Non-linear programming method for multi-criteria decision making problems under interval neutrosophic set environment
Interval neutrosophic sets (INSs) capturing the uncertainties by characterizing into the intervals of the truth, the indeterminacy, and the falsity membership degrees, is a more flexible way to explore the decision-making applications. In this paper, we develop a nonlinear programming (NP) model based on the technique for order preference by similarity to ideal solution (TOPSIS), to solve decision-making problems in which criterion values and their importance are given in the form of interval neutrosophic numbers (INNs). Based on the concept of closeness coefficient, we firstly construct a pair of the nonlinear fractional programming model and then transform it into the linear programming model. Furthermore, to determine the ranking of considered alternatives, likelihood-based comparison relations are constructed. Finally, an illustrative example demonstrates the applicability of the proposed method for dealing with decision making problems with incomplete knowledge.
Journal Article
Convex mixed-integer nonlinear programs derived from generalized disjunctive programming using cones
We propose the formulation of convex Generalized Disjunctive Programming (GDP) problems using conic inequalities leading to conic GDP problems. We then show the reformulation of conic GDPs into Mixed-Integer Conic Programming (MICP) problems through both the big-M and hull reformulations. These reformulations have the advantage that they are representable using the same cones as the original conic GDP. In the case of the hull reformulation, they require no approximation of the perspective function. Moreover, the MICP problems derived can be solved by specialized conic solvers and offer a natural extended formulation amenable to both conic and gradient-based solvers. We present the closed form of several convex functions and their respective perspectives in conic sets, allowing users to formulate their conic GDP problems easily. We finally implement a large set of conic GDP examples and solve them via the scalar nonlinear and conic mixed-integer reformulations. These examples include applications from Process Systems Engineering, Machine learning, and randomly generated instances. Our results show that the conic structure can be exploited to solve these challenging MICP problems more efficiently. Our main contribution is providing the reformulations, examples, and computational results that support the claim that taking advantage of conic formulations of convex GDP instead of their nonlinear algebraic descriptions can lead to a more efficient solution to these problems.
Journal Article
How to convexify the intersection of a second order cone and a nonconvex quadratic
A recent series of papers has examined the extension of disjunctive-programming techniques to mixed-integer second-order-cone programming. For example, it has been shown—by several authors using different techniques—that the convex hull of the intersection of an ellipsoid,
E
, and a split disjunction,
(
l
-
x
j
)
(
x
j
-
u
)
≤
0
with
l
<
u
, equals the intersection of
E
with an additional second-order-cone representable (SOCr) set. In this paper, we study more general intersections of the form
K
∩
Q
and
K
∩
Q
∩
H
, where
K
is a SOCr cone,
Q
is a nonconvex cone defined by a single homogeneous quadratic, and
H
is an affine hyperplane. Under several easy-to-verify conditions, we derive simple, computable convex relaxations
K
∩
S
and
K
∩
S
∩
H
, where
S
is a SOCr cone. Under further conditions, we prove that these two sets capture precisely the corresponding conic/convex hulls. Our approach unifies and extends previous results, and we illustrate its applicability and generality with many examples.
Journal Article