Separable boolean functions and generalized Fibonacci sequences

It is well known that the Boolean functions can be represented by two-layer perceptrons, and a part of them, namely separable Boolean functions, can be represented by one-layer perceptrons. How many separable Boolean functions of n variables there are is an open problem. On the other hand, given a n-element set X, how many antichains does P(X) have is also an open problem. This paper established an inequality reflecting the relationship between these two open problems. Second, this paper introduced two classes of Boolean functions which are generalizations of AND-OR functions and OR-AND functions, respectively, and proved that they are all separable and the weights in representing them are exactly terms of corresponding generalized Fibonacci sequences.     

Verification of discrete time stochastic hybrid systems: A stochastic reach-avoid decision problem

We present a dynamic programming based solution to a probabilistic reach-avoid problem for a controlled discrete time stochastic hybrid system. We address two distinct interpretations of the reach-avoid problem via stochastic optimal control. In the first case, a sum-multiplicative cost function is introduced along with a corresponding dynamic recursion which quantifies the probability of hitting a target set at some point during a finite time horizon, while avoiding an unsafe set during each time step preceding the target hitting time. In the second case, we introduce a multiplicative cost function and a dynamic recursion which quantifies the probability of hitting a target set at the terminal time, while avoiding an unsafe set during the preceding time steps. In each case, optimal reach while avoid control policies are derived as the solution to an optimal control problem via dynamic programming. Computational examples motivated by two practical problems in the management of fisheries and finance are provided.     

A new approach to differential dynamic programming for discrete time systems

This paper proposes a new differential dynamic programming algorithm for solving discrete time optimal control problems with equality and inequality constraints on both control and state variables and proves its convergence. The present algorithm is different from differential dynamic programming algorithms developed in [10]-[15], which can hardly solve optimal control problems with inequality constraints on state variables and whose convergence has not been proved. Composed of iterative methods for solving systems of nonlinear equations, it is based upon Kuhn-Tucker conditions for recurrence relations of dynamic programming. Numerical examples show file efficiency of the present algorithm.     

A linear programming approach to max-sum problem: a review

The max-sum labeling problem, defined as maximizing a sum of binary (i.e., pairwise) functions of discrete variables, is a general NP-hard optimization problem with many applications, such as computing the MAP configuration of a Markov random field. We review a not widely known approach to the problem, developed by Ukrainian researchers Schlesinger et al. in 1976, and show how it contributes to recent results, most importantly, those on the convex combination of trees and tree-reweighted max-product. In particular, we review Schlesinger et al.'s upper bound on the max-sum criterion, its minimization by equivalent transformations, its relation to the constraint satisfaction problem, the fact that this minimization is dual to a linear programming relaxation of the original problem, and the three kinds of consistency necessary for optimality of the upper bound. We revisit problems with Boolean variables and supermodular problems. We describe two algorithms for decreasing the upper bound. We present an example application for structural image analysis.     

Application of mathematical programming in the design of optimal control systems†

The problem of applying the various computational methods of mathematical programming in the design of an optimal control system is discussed. A general case of non-linear, non-autonomous, state equations, subject to inequality constraints on both state and control variables, is considered. Both continuous and discrete time systems are investigated. In case of discrete time systems, the sampling intervals are assumed generally unequal and aperiodic, with inequality constraints imposed upon them.

Systems like these impose considerable computational difficulties when treated by the maximum principle or dynamie programming. Using mathematical programming, one may simplify a wide class of those computational problems.

Several examples of applying mathematical programming to particular control problems are presented.     

Application of interval predictor model into robust model predictive control

Here we apply interval prediction model into robust model predictive control (MPC) strategy. After introducing the family of models and some basic information, we present the computational results for the construction of interval predictor model, whose regression parameters are included in a sphere parameter set. Given a size measure to scale the average amplitude of the predictor interval, one optimal model that minimises a size measure is efficiently computed by solving a linear programming problem. We apply the active set approach to solve the linear programming problem and based on these optimisation variables, the predictor interval of the considered model with sphere parameter set can be constructed. As for a fixed non-negative number from the size measure, we propose a better choice by using the optimality conditions. In order to apply interval prediction model into robust MPC, two strategies are proposed to analyse a min-max optimisation problem. After input and output variables are regarded as decision variables, a standard quadratic optimisation is obtained and its dual form is derived, then Gauss–Seidel algorithm is proposed to solve the dual problem and convergence of Gauss–Seidel algorithm is provided too. Finally two simulation examples confirm the theoretical results.     

Some practical experiences in applying nonlinear programming to CAD

Nonlinear programming is one method of solving process design problems. The formulation of a problem is: given a revenue or cost function subject to both equality and inequality constraints, find the optimal vector of design variables or operating conditions. Difficulties in meshing codes that solve nonlinear programming problems with process models and simulation packages are discussed here.Major difficulties arise because the representation of operations in a chemical plant involves highly modular equations. subroutines from which the equations cannot be readily extracted, implicit relations between subroutines and associated databases, and large numbers of constraints and variables.In addition, certain problems arise because of the iterative numerical techniques of solution employed in the optimization problems such as: how can suitable initial guesses for the independent variables be obtained, how can numerical computational errors be reduced, how can ill-conditioning of the matrices used be diminished, and how can undefined arguments of the constraints and objective function be avoided. All of these factors and ways to resolve them are considered.We also discuss briefly the relative performance of nonlinear programming codes that can be employed in design.     

A quasilinearization algorithm and its application to a manipulator problem

A quasilinearization algorithm is proposed for the computation of optimal control of a class of constrained problem. The constraints are inequality constraints on functions of the state and control variables, and bounds on the values of the control variables. Necessary conditions for optimal control of the control problem are derived. In the iterative procedure, no prior information is required regarding the sequence of constrained and unconstrained arcs and the inequality constraints which are on their boundaries along a specific constrained arc of the optimal trajectory. All this information will be determined within the iterative procedure using some necessary conditions for optimal control. The ability of the proposed algorithm to solve practical problems is demonstrated by its application to several variations of two problems, one of which is a common manipulator problem in industry where transportation of open vessels of liquid is to be performed in a specified period of time. It is shown that the proposed quasilinearization algorithm is an effective tool in deriving optimal control policies for a common type of manipulator operation in industry.     

Constrained Optimal Hybrid Control of a Flow Shop System

We consider an optimal control problem for the hybrid model of a deterministic flow shop system, in which the jobs are processed in the order they arrive at the system. The problem is decomposed into a higher-level discrete-event system control problem of determining the optimal service times, and a set of lower-level classical control problems of determining the optimal control inputs for given service times. We focus on the higher-level problem which is nonconvex and nondifferentiable. The arrival times are known and the decision variables are the service times that are controllable within constraints. We present an equivalent convex optimization problem with linear constraints. Under some cost assumptions, we show that no waiting is observed on the optimal sample path. This property allows us to simplify the convex optimization problem by eliminating variables and constraints. We also prove, under an additional strict convexity assumption, the uniqueness of the optimal solution and propose two algorithms to decompose the simplified convex optimization problem into a set of smaller convex optimization problems. The effects of the simplification and the decomposition on the solution times are shown on an example problem.     

Exact and heuristic methods for solving Boolean games

Boolean games are a framework for reasoning about the rational behavior of agents whose goals are formalized using propositional formulas. Compared to normal form games, a well-studied and related game framework, Boolean games allow for an intuitive and more compact representation of the agents’ goals. So far, Boolean games have been mainly studied in the literature from the Knowledge Representation perspective, and less attention has been paid on the algorithmic issues underlying the computation of solution concepts. Although some suggestions for solving specific classes of Boolean games have been made in the literature, there is currently no work available on the practical performance. In this paper, we propose the first technique to solve general Boolean games that does not require an exponential translation to normal-form games. Our method is based on disjunctive answer set programming and computes solutions (equilibria) of arbitrary Boolean games. It can be applied to a wide variety of solution concepts, and can naturally deal with extensions of Boolean games such as constraints and costs. We present detailed experimental results in which we compare the proposed method against a number of existing methods for solving specific classes of Boolean games, as well as adaptations of methods that were initially designed for normal-form games. We found that the heuristic methods that do not require all payoff matrix entries performed well for smaller Boolean games, while our ASP based technique is faster when the problem instances have a higher number of agents or action variables.     

Exact solutions of some optimization problems of transport logistics

A problem for constructing the shortest cyclic route that ensures homogeneous cargo is delivered from producers to consumers by a transport vehicle of limited capacity is considered. Formalizations in the form of Boolean quadratic programming and linear integer programming problems are given. Comparative analysis of the efficiency of three exact algorithms is performed. The problem of finding the minimal admissible capacity of the transport vehicle is considered as an auxiliary problem. Dependence of the length of the optimal route on the capacity of the transport vehicle is studied experimentally.     

An integer programming approach to optimal control problems in context-sensitive probabilistic Boolean networks

A Boolean network is one of the models of biological networks such as gene regulatory networks, and has been extensively studied. In particular, a probabilistic Boolean network (PBN) is well known as an extension of Boolean networks, but in the existing methods to solve the optimal control problem of PBNs, it is necessary to compute the state transition diagram with 2ⁿ nodes for a given PBN with n states. To avoid this computation, an integer programming-based approach is proposed for a context-sensitive PBN (CS-PBN), which is a general form of PBNs. In the proposed method, a CS-PBN is transformed into a linear system with binary variables, and the optimal control problem is reduced to an integer linear programming problem. By a numerical example, the effectiveness of the proposed method is shown.     

Using variable reduction strategy to accelerate evolutionary optimization

In this study, we introduce a novel approach of variable reduction and integrate it into evolutionary algorithms in order to reduce the complexity of optimization problems. We develop reduction processes of variable reduction for derivative unconstrained optimization problems (DUOPs) and constrained optimization problems (COPs) with equality constraints and active inequality constraints. Variable reduction uses the problem domain knowledge implied when investigating optimal conditions existing in optimization problems. For DUOPs, equations involving derivatives are considered while for COPs, we discuss equations expressing the equality constraints. From the relationships formed in this way, we obtain relationships among the variables that have to be satisfied by optimal solutions. According to such relationships, we can utilize some variables (referred to as core variables) to express some other variables (referred to as reduced variables). We show that the essence of variable reduction is to produce a minimum collection of core variables and a maximum number of reduced variables based on a system of equations. We summarize some application-oriented situations of variable reduction and stress several important issues related to the further application and development of variable reduction. Essentially, variable reduction can reduce the number of variables and eliminate equality constraints, thus reducing the dimensionality of the solution space and improving the efficiency of evolutionary algorithms. The approach can be applied to unconstrained, constrained, continuous and discrete optimization problems only if there are explicit variable relationships to be satisfied in the optimal conditions. We test variable reduction on real-world and synthesized DUOPs and COPs. Experimental results and comparative studies point at the effectiveness of variable reduction.     

General redundancy optimization method for cooperating manipulatorsusing quadratic inequality constraints

In redundancy optimization problems related to cooperating manipulators such as optimal force distribution, constraints on the physical limits of the manipulators should be considered. We propose quadratic inequality constraints (QICs), which lead to ellipsoidal feasible regions, to solve the optimization problem more efficiently. We investigate the effect of the use of QICs from the points of view of problem size and change of the feasible region. To efficiently deal with the QICs, we also propose the dual quadratically constrained quadratic programming (QCQP) method. In this method, the size of the optimization problem is reduced so that the computational burden is lightened. The proposed method and another well-known quadratic programming method are applied to the two PUMA robots system and compared with each other. The results show that the use of QICs with the dual QCQP method allows for faster computation than the existing method     

On reducing a quantile optimization problem with discrete distribution to a mixed integer programming problem

We propose an equivalent reduction of the quantile optimization problem with a discrete distribution of random parameters to a partially integer programming problem of large dimension. The number of integer (Boolean) variables in this problem equals the number of possible values for the random parameters vector. The resulting problems can be solved with standard discrete optimization software. We consider applications to quantile optimization of a financial portfolio and show results of numerical experiments.     

具有不等式路径约束的微分代数方程系统的动态优化

针对具有不等式路径约束的微分代数方程（Differential-algebraic equations，DAE）系统的动态优化问题，通常将DAE中的等式路径约束进行微分处理，或者将其转化为点约束或不等式约束进行求解.前者需要考虑初值条件的相容性或增加约束，在变量间耦合度较高的情况下这种转化求解方法是不可行的；后者将等式约束转化为其他类型的约束会增加约束条件，增加了求解难度.为了克服该缺点，本文提出了结合后向差分法对DAE直接处理来求解上述动态优化问题的方法.首先利用控制向量参数化方法将无限维的最优控制问题转化为有限维的最优控制问题，再利用分点离散法用有限个内点约束去代替原不等式路径约束，最后用序列二次规划（Sequential quadratic programming，SQP）法使得在有限步数的迭代下，得到满足用户指定的路径约束违反容忍度下的KKT（Karush Kuhn Tucker）最优点.理论上证明了该算法在有限步内收敛.最后将所提出的方法应用在具有不等式路径约束的微分代数方程系统中进行仿真，结果验证了该方法的有效性.     

一种新的非线性规划神经网络模型

提出一种新型的求解非线性规划问题的神经网络模型.该模型由变量神经元、Lagrange 乘子神经元和Kuhn-Tucker乘子神经元相互连接构成.通过将Kuhn-Tucker乘子神经元限 制在单边饱和工作方式,使得在处理非线性规划问题中不等式约束时不需要引入松弛变量,避 免了由于引入松弛变量而造成神经元数目的增加,有利于神经网络的硬件实现和提高神经网 络的收敛速度.可以证明,在适当的条件下,文中提出的神经网络模型的状态轨迹收敛到与非 线性规划问题的最优解相对应的平衡点.     

具有未知参数的非线性系统动态优化

针对具有未知参数和不等式路径约束的非线性系统动态优化问题,提出一种新颖有效的数值求解方法.首先,将未知参数视为一个动态优化问题的决策变量;其次,利用多重打靶法将无限维的含未知参数动态优化问题转化为有限维的非线性规划问题,进而在不等式路径约束违反的时间段内,用有限多个内点约束替代原不等式路径约束;然后,用内点法求解转化后的非线性规划问题,在路径约束违反的一定容许度下,经过有限多次步数迭代后得到未知参数值的同时得到控制策略,并在理论上对所提出算法的收敛性进行相应证明;最后,对两个经典的含未知参数非线性系统的动态优化问题进行数值仿真以验证所提出算法的有效性.     

Controller design with multiple objectives

In this paper we study multi-objective control problems that give rise to equivalent convex optimization problems. We develop a uniform treatment of such problems by showing their equivalence to linear programming problems with equality constraints and an appropriate positive cone. We present some specialized results on duality theory, and we apply them to the study of three multi-objective control problems: the optimal l₁ control with time-domain constraints on the response to some fixed input, the mixed H₂/l₁ -control problem, and the l₁ control with magnitude constraint on the frequency response. What makes these problems complicated is that they are often equivalent to infinite-dimensional optimization problems. The characterization of the duality relationship between the primal and dual problem allows us to derive several results. These results establish connections with special convex problems (linear programming or linear matrix inequality problems), uncover finite-dimensional structures in the optimal solution, when possible, and provide finite-dimensional approximations to any degree of accuracy when the problem does not appear to have a finite-dimensional structure. To illustrate the theory and highlight its potential, several numerical examples are presented     

Reducing Two-Stage Probabilistic Optimization Problems with Discrete Distribution of Random Data to Mixed-Integer Programming Problems*

We consider two-stage stochastic programming models with quantile criterion as well as models with a probabilistic constraint on the random values of the objective function of the second stage. These models allow us to formalize the requirements for the reliability and safety of the system being optimized and to optimize system’s performance under extreme conditions. We propose a method of equivalent transformation of these models under discrete distribution of random parameters to mixed-integer programming problems. The number of additional integer (Boolean) variables in these problems equals to the number of possible values of the vector of random parameters. The obtained mixed optimization problems can be solved by powerful standard discrete optimization software. To illustrate the approach, the results of numerical experiment for the problem of small dimension are presented.