Equivalent Nonlinear Programming Research Articles

A stochastic technique to solve interval non-linear programming problems using GH-difference

PurposeA stochastic technique for solving interval non-linear problems using generalized Hukuhara (GH)-difference is proposed. The non-linear programming problem in interval form is transformed into an equivalent non-linear programming problem with real coefficients by associating a Gaussian random variable to the interval and the six-sigma rule. The conceptualized idea eliminates the decision maker’s instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. To demonstrate a coherent understanding, numerical examples have been used.Design/methodology/approachA stochastic approach has been used to develop a solution technique for solving interval nonlinear programming problems which arise in the modeling of scientific and engineering problems under uncertain environments.FindingsThe proposed idea eliminates the decision maker’s instinctive selection of weight functions and provides an alternative to the order relation method, max-min criteria-based methods and bi-level approaches for representing intervals as real numbers. This method provides specific results rather than in the interval form, which are more practical and implementable by the decision maker.Originality/valueThis is to certify, that the research paper submitted is an outcome of original work. I have duly acknowledged all the sources from which the ideas and extracts have been taken. This article has not been submitted elsewhere for publication.

Can we compute the worst voltage case under power flows uncertainty at TSO-DSO interfaces?

This paper proposes the new problem of calculating the worst-voltage-case (WVC) under power flows uncertainty, particularly at interfaces between transmission and distribution grid operators. The goal of the calculation is to verify if the realization of the worst uncertainty scenario leads to abnormal voltages, or assure where the voltage instability is not a risk, and inform the transmission system operator. The problem is formulated as a tailored non-convex, nonlinear optimal power flow (OPF) with complementarity or equilibrium constraints (ECs). The latter model the generator switch between under voltage control and under reactive power limit. The problem is converted into a mathematically equivalent nonlinear programming OPF problem in which ECs are handled as penalty terms in the objective function. The paper has conducted a series of experiments on the 60-bus Nordic system, which have pointed out the challenges associated to such computations to obtain meaningful results and unveiled new insights. The findings and scalability are confirmed using a real-world 1203-bus system.

A novel variational inequality approach for modeling the optimal equilibrium in multi-tiered supply chain networks

We present a novel variational inequality model (VIM) to capture the complex real decision-making process in multi-tiered supply chain networks (MSCN) without strictly limiting the features of related functions. The VIM is formulated with the equilibrium conditions on links as the optimization goal and the flow conservation condition as the main constraints. We transform the VIM into a series of equivalent Non-Linear Programming Models (NLPMs) to solve. To address this challenge, we propose a novel population-based heuristic algorithm called the Multiscale Model Learning Algorithm (MMLA). The MMLA is inspired by the learning behavior of individuals in a group and can converge to an optimal equilibrium state of the MSCN. The MMLA has two key operations: zooming in on the search field and learning search in a learning stage. The excellent performers, called medalists, are imitated by other learners. With the increase in learning stages, the learning efficiency is improved, and the searching energy is concentrated in a more promising area. We employ sixteen benchmark optimization problems and two supply chain networks to demonstrate the effectiveness of the MMLA and the rationality of the equilibrium models. The results obtained by MMLA for the NLPM show that the MMLA can solve the equilibrium model effectively, and multiple optimal equilibrium states may exist for an MSCN. The flexibility of the NLPM makes it possible to consider more complicated decision-making mechanisms in the model.

Energy-efficient predictive control for trams incorporating disjunctive time constraints from traffic lights

Tram operations are often blocked by traffic lights, leading to frequent decelerations and re-accelerations that increase operational energy consumption. This paper focuses on tram energy-efficient control problem incorporating time constraints from traffic lights that have multiple feasible green time windows (GTWs). We formulate the problem as a mixed-integer nonlinear program (MINLP), where binary variables are assigned to model disjunctive time constraints of the GTWs. To address computational challenge of solving the MINLP, we reformulate it as a tractable nonlinear program (NLP). Specifically, an equivalent NLP is first presented by replacing the integrality constraint with nonlinear constraints, and then the nonlinear constraints are relaxed and penalized into cost functions. To recover a solution of the MINLP, we propose a computationally efficient sequential quadratic programming algorithm in a shrinking horizon model predictive control framework, which updates the penalty parameter and quadratic programming subproblems in parallel. The solution obtained from the subproblem is feasible in each iteration, and convergence of the feasibility iterations can be enforced by the updated penalty. The performance of the proposed approach is investigated on different scenarios using real-life tram data. Results show that the method is able to generate energy-efficient driving trajectories in a dynamic environment, while crossing traffic lights in effective GTWs without unnecessary decelerations and re-accelerations.

Global optimality in minimum compliance topology optimization of frames and shells by moment-sum-of-squares hierarchy

The design of minimum-compliance bending-resistant structures with continuous cross-section parameters is a challenging task because of its inherent non-convexity. Our contribution develops a strategy that facilitates computing all guaranteed globally optimal solutions for frame and shell structures under multiple load cases and self-weight. To this purpose, we exploit the fact that the stiffness matrix is usually a polynomial function of design variables, allowing us to build an equivalent non-linear semidefinite programming formulation over a semi-algebraic feasible set. This formulation is subsequently solved using the Lasserre moment-sum-of-squares hierarchy, generating a sequence of outer convex approximations that monotonically converges from below to the optimum of the original problem. Globally optimal solutions can subsequently be extracted using the Curto-Fialkow flat extension theorem. Furthermore, we show that a simple correction to the solutions of the relaxed problems establishes a feasible upper bound, thereby deriving a simple sufficient condition of global $\varepsilon$-optimality. When the original problem possesses a unique minimum, we show that this solution is found with a zero optimality gap in the limit. These theoretical findings are illustrated on several examples of topology optimization of frames and shells, for which we observe that the hierarchy converges in a finite (rather small) number of steps.

Time-optimal control problem for a linear parameter varying system with nonlinear item

In this paper, we considered a time-optimal control problem for a new type of linear parameter varying (LPV) system which is obtained through data identification in the process of dealing with actual problems. The addition of non-linear terms is compensation for the method that does not require linear expansion at the equilibrium point. Since the objective function is the terminal time which is an implicit function concerning decision variables, it is a non-standard optimal control problem with uncertain terminal time. To find the global optimal solution to this problem, firstly, the control parameterization method is used to transform it into a nonlinear optimization problem of parameter selection, and then the modifed particle swarm optimization (PSO) algorithm is combined to solve the equivalent nonlinear programming problem. Numerical examples are used to illustrate the effectiveness of the proposed algorithm.

Approaches for Solving Fully Fuzzy Rough Multi-Objective Nonlinear Programming Problems

Practical nonlinear programming problem often encounters uncertainty and indecision due to various factors that cannot be controlled. To overcome these limitations, fully fuzzy rough approaches are applied to such a problem. In this paper, an effective two approaches are proposed to solve fully fuzzy rough multi-objective nonlinear programming problems (FFRMONLP) where all the variables and parameters are fuzzy rough triangular numbers. The first, based on a slice sum technique, a fully fuzzy rough multi-objective nonlinear problem has turned into five equivalent multi-objective nonlinear programming (FFMONLP) problems. The second proposed method for solving FFRMONLP problems is α-cut approach, where the triangular fuzzy rough variables and parameters of the FFRMONLP problem are converted into rough interval variables and parameters by α-level cut, moreover the rough MONLP problem turns into four MONLP problems. Furthermore, the weighted sum method is used in both proposed approaches to convert multi-objective nonlinear problems into an equivalent nonlinear programming problem. Finally, the effectiveness of the proposed procedure is demonstrated by numerical examples.

A novel two-phase approach for the bi-objective simultaneous delivery and pickup problem with fuzzy pickup demands

This paper is dedicated to providing a solution framework for the bi-objective vehicle routing problem with simultaneous delivery and pickup which aims to minimize the comprehensive cost as well as maximize the recycling revenue in each round of dispatching. Considering that the real weights of goods to be recycled from the customers are fixed but cannot be precisely given while making one-time routing scheme in the daily operations, the pickup demands are considered as fuzzy numbers, and accordingly a fuzzy chance-constraint programming model for obtaining a prior route solution from the perspective of risk is presented. Afterwards, by demonstrating the continuity and monotonicity of the objective functions, a two-phase approach based on the operational law of the inverse credibility distribution is introduced to solve the model, including translating it into an equivalent nonlinear programming model and resorting to the existing algorithms for obtaining optimal solutions afterwards. Subsequently, in order to validate the performance of the proposed approach and in consideration of the inherent complexity of the vehicle routing problem, a two-phase-based genetic algorithm and the conventional fuzzy simulation-based genetic algorithm are designed and compared by a clothes delivery and pickup problem. The computational results demonstrate that the proposed solution framework is competitive in effectiveness and efficiency, and the parameter analyses provide some suggestions for guidance.

Solving network-constrained nonsmooth economic dispatch problems through a gradient-based approach

The Network-Constrained Nonsmooth Economic Dispatch (NCNS-ED) is formulated as a nonsmooth, nonconvex and multimodal optimization problem. Nonsmoothness is generally associated with valve-point loading effects in the cost function of thermal units. Due to nonsmoothness, gradient-based approaches cannot be used directly for solving NCNS-ED problems. Hence, heuristic approaches have been used as the main methodologies for solving this problem. In this paper, we propose a reformulation for the NCNS-ED problem that enable its solution by means of gradient-based approaches. This formulation is based on the definition of an equivalent nonlinear programming problem where the modular terms are eliminated by using artificial variables and inequality constraints. We propose a primal–dual modified log-barrier function approach for solving the reformulated problem, where each inequality constraint is penalized by using a piecewise function that involves the modified barrier functions of Polyak and Jittortrum-Osborne-Meggido. Numerical results involving small-, medium- and large-scale power systems, have shown that the proposed reformulation enables eliminating nonsmoothness in the NCNS-ED problems. The method proposed, which is compared to some methods described the literature, has shown robustness for solving real large-scale NCNS-ED problems with acceptable computation times.

A new approach for determining multi-objective optimal control of semilinear parabolic problems

In this paper, two approaches based on evolutionary algorithms are applied to solve a multi-objective optimal control problem governed by semilinear parabolic partial differential equations. In this approach, first, we change the problem into a measure-theoretical one, replace this with an equivalent infinite dimensional multi-objective nonlinear programming problem and apply approximating schemes. Finally, non-dominated sorting genetic algorithm and multi-objective particle swarm optimization are employed to obtain Pareto optimal solutions of the problem. Numerical examples are presented to show the efficiency of the given approach.

Linear fractional programming problems with some multi-choice parameters

Linear fractional programming is a class of mathematical programming problem where we optimise the ratio of two linear functions subject to some linear constraints. In this paper, we present a linear fractional programming model where some or all the parameters are multi-choice type. We present a novel and efficient method, which integrates classical Charnes-Cooper transformation and Lagrange's interpolating polynomial, to transform multi-choice linear fractional programming problems into an equivalent mixed-integer nonlinear programming (MINLP) problems. A theorem is presented to establish the relation between the optimal solution of the multi-choice linear fractional programs and the equivalent MINLP. Some numerical examples are studied to illustrate the methodology.

A New Direct Method for Solving Optimal Control Problem of Nonlinear Volterra–Fredholm Integral Equation via the Müntz–Legendre Polynomials

In this study, a new pseudo-spectral method is proposed to numerical solve of an optimal control problem whose constraint is given in terms of integral equation. The computational technique is based on a special form of the orthogonal Muntz–Legendre polynomials. Using the Legendre–Gauss–Lobatto points and Legendre–Gauss–Lobatto quadrature, the control and state functions are approximated by a finite combination of basis functions. Employing the method of this paper, the main optimal control problem becomes to an equivalent nonlinear programming problem. We also analyze the convergence of the proposed method by applying appropriate conditions. Finally, some examples are included to demonstrate the performance and accuracy of the introduced scheme.

Convergence of a Scholtes-type regularization method for cardinality-constrained optimization problems with an application in sparse robust portfolio optimization

We consider general nonlinear programming problems with cardinality constraints. By relaxing the binary variables which appear in the natural mixed-integer programming formulation, we obtain an almost equivalent nonlinear programming problem, which is thus still difficult to solve. Therefore, we apply a Scholtes-type regularization method to obtain a sequence of easier to solve problems and investigate the convergence of the obtained KKT points. We show that such a sequence converges to an S-stationary point, which corresponds to a local minimizer of the original problem under the assumption of convexity. Additionally, we consider portfolio optimization problems where we minimize a risk measure under a cardinality constraint on the portfolio. Various risk measures are considered, in particular Value-at-Risk and Conditional Value-at-Risk under normal distribution of returns and their robust counterparts under moment conditions. For these investment problems formulated as nonlinear programming problems with cardinality constraints we perform a numerical study on a large number of simulated instances taken from the literature and illuminate the computational performance of the Scholtes-type regularization method in comparison to other considered solution approaches: a mixed-integer solver, a direct continuous reformulation solver and the Kanzow–Schwartz regularization method, which has already been applied to Markowitz portfolio problems.

Distributionally robust fuzzy project portfolio optimization problem with interactive returns

Effective project selection and staff assignment strategies directly impact organizational profitability. Based on critical value optimization criterion, this paper discusses how uncertainty and interaction impact the project portfolio return and staff allocation. Since the exact possibility distributions of uncertain parameters in practical project portfolio problems are often unavailable, we adopt variable parametric possibility distributions to characterize uncertain model parameters. Furthermore, this paper develops a novel parametric credibilistic optimization method for project portfolio selection problem. According to the structural characteristics of variable parametric possibility distributions, we derive the equivalent analytical expressions of credibility constraints, and turn the original credibilistic project portfolio model into its equivalent nonlinear mixed-integer programming models. To show the advantages of the proposed parametric credibilistic optimization method, some numerical experiments are conducted by setting various values of distribution parameters. The computational results support our arguments by comparing with the optimization method under fixed possibility distributions.

Constructing the Pareto front for multi-objective Markov chains handling a strong Pareto policy approach

In this paper, we present a multi-objective solution of the Pareto front for a certain class of discrete-time ergodic controllable Markov chains. For solving the problem, we transform the original multi-objective problem into an equivalent nonlinear programming problem implementing the Lagrange principle. We then propose to adopt the Tikhonov’s regularization method to ensure the convergence of the cost functions to a unique point into the Pareto front. We show that Pareto policies are characterized as optimal policies. One of the fundamental problems related to the construction of the Pareto front is the existence and characterization of both, Pareto and strong Pareto policies. By introducing the Tikhonov’s regularizator, we ensure the existence of strong Pareto policies. This paper proposes a real solution to this problem. We formulate the original problem considering several constraints: (a) we employ the c-variable method for the introduction of linear constraints over the nonlinear problem and (b) restrict the cost functions, allowing points in the Pareto front to have a small distance from one another. The constraints imposed by the c-variable method make the problem computationally tractable and the restriction imposed by the small distance change ensures the continuation of the Pareto front. The resulting equation in this nonlinear system is an optimization problem for which the necessary and efficient condition of a minimum is solved using the projected gradient method. Moreover, we also prove the convergence conditions and compute the estimate rate of convergence of variables corresponding to the Lagrange principle and the Tikhonov’s regularization, respectively. We provide all the details needed to implement the proposed method in an efficient way. The usefulness of the method is successfully demonstrated by a numerical example.

Necessary and sufficient Karush–Kuhn–Tucker conditions for multiobjective Markov chains optimality

The solution concepts proposed in this paper follow the Karush–Kuhn–Tucker (KKT) conditions for a Pareto optimal solution in finite-time, ergodic and controllable Markov chains multi-objective programming problems. In order to solve the problem we introduce the Tikhonov’s regularizator for ensuring the objective function is strict-convex. Then, we consider the c-variable method for introducing equality constraints that guarantee the result belongs to the simplex and satisfies ergodicity constraints. Lastly, we restrict the cost-functions allowing points in the Pareto front to have a small distance from one another. The computed image points give a continuous approximation of the whole Pareto surface. The constraints imposed by the c-variable method make the problem computationally tractable and, the restriction imposed by the small distance change ensures the continuation of the Pareto front. We transform the multi-objective nonlinear problem into an equivalent nonlinear programming problem by introducing the Lagrange function multipliers. As a result we obtain that the objective function is strict-convex, the inequality constraints are continuously differentiable and the equality constraint is an affine function. Under these settings, the KKT optimality necessary and sufficient conditions are elicited naturally. A numerical example is solved for providing the basic techniques to compute the Pareto optimal solutions by resorting to KKT conditions.

Energy Efficiency Resource Allocation for MIMO Cognitive Radio with Multiple Antenna Spectrum Sensing

The energy-efficient design of sensing-based spectrum sharing of a multi-input and multi-output (MIMO) cognitive radio (CR) system with imperfect multiple antenna spectrum sensing is investigated in this study. Optimal resource allocation strategies, including sensing time and power allocation schemes, are studied to maximize the energy efficiency (EE) of the secondary base station under the transmit power and interference power constraints. EE problem is formulated as a nonlinear stochastic fractional programming of a nonconvex optimal problem. The EE problem is transformed into its equivalent nonlinear parametric programming and solved by one-dimension search algorithm. To reduce searching complexity, the search range was founded by demonstration. Furthermore, simulation results confirms that an optimal sensing time exists to maximize EE, and shows that EE is affected by the spectrum detection factors and corresponding constraints.

Joint Stocking and Product Offer Decisions Under the Multinomial Logit Model

This article studies a joint stocking and product offer problem. We have access to a number of products to satisfy the demand over a finite selling horizon. Given that customers choose among the set of offered products according to the multinomial logit model, we need to decide which sets of products to offer over the selling horizon and how many units of each product to stock so as to maximize the expected profit. We formulate the problem as a nonlinear program, where the decision variables correspond to the stocking quantity for each product and the duration of time that each set of products is offered. This nonlinear program is intractable due to its large number of decision variables and its nonseparable and nonconcave objective function. We use the structure of the multinomial logit model to formulate an equivalent nonlinear program, where the number of decision variables is manageable and the objective function is separable. Exploiting separability, we solve the equivalent nonlinear program through a dynamic program with a two dimensional and continuous state variable. As the solution of the dynamic program requires discretizing the state variable, we study other approximate solution methods. Our equivalent nonlinear program and approximate solution methods yield insights for good offer sets.

A Fuzzy Nonlinear Programming Approach for Optimizing the Performance of a Four-Objective Fluctuation Smoothing Rule in a Wafer Fabrication Factory

In theory, a scheduling problem can be formulated as a mathematical programming problem. In practice, dispatching rules are considered to be a more practical method of scheduling. However, the combination of mathematical programming and fuzzy dispatching rule has rarely been discussed in the literature. In this study, a fuzzy nonlinear programming (FNLP) approach is proposed for optimizing the scheduling performance of a four-factor fluctuation smoothing rule in a wafer fabrication factory. The proposed methodology considers the uncertainty in the remaining cycle time of a job and optimizes a fuzzy four-factor fluctuation-smoothing rule to sequence the jobs in front of each machine. The fuzzy four-factor fluctuation-smoothing rule has five adjustable parameters, the optimization of which results in an FNLP problem. The FNLP problem can be converted into an equivalent nonlinear programming (NLP) problem to be solved. The performance of the proposed methodology has been evaluated with a series of production simulation experiments; these experiments provide sufficient evidence to support the advantages of the proposed method over some existing scheduling methods.

A Fuzzy Collaborative Forecasting Approach for Forecasting the Productivity of a Factory

Productivity is always considered as one of the most basic and important factors to the competitiveness of a factory. For this reason, all factories have sought to enhance productivity. To achieve this goal, we first need to estimate the productivity. However, there is considerable degree of uncertainty in productivity. For this reason, a fuzzy collaborative forecasting approach is proposed in this study to forecast the productivity of a factory. First, a learning model is established to estimate the future productivity. Subsequently, the learning model is converted into three equivalent nonlinear programming problems to be solved from various viewpoints. The fuzzy productivity forecasts by different experts may not be equal and should therefore be aggregated. To this end, the fuzzy intersection and back propagation network approach is applied. The practical example of a dynamic random access memory (DRAM) factory is used to evaluate the effectiveness of the proposed methodology.