General Objective Function Research Articles

A two-level vertex-searching global algorithm framework for bilevel linear fractional programming problems

ABSTRACT We propose a new two-level vertex-searching algorithm framework that finds a global optimal solution to the continuous bilevel linear fractional programming problem over a compact polyhedron, in which both the upper and the lower objectives are linear fractional. Our solution method adopts the vertex-searching approach on the polyhedron, and the search space is determined by the set of candidates of optimal base to the lower level problem. In order to search base, a modified enumerative scheme, that is a new upper bound filter scheme inserted into the classical enumerative scheme, is proposed. The main solution procedure is designed on solving a sequence of upper and lower level mathematical programs; instead of a single-level problem reformulation approach, which is popularly and widely used in literature. An extension on general upper level objective functions such as quasiconvex/quasiconcave for the proposed vertex-searching approach is discussed. Numerical experiments show that our algorithm leads us to a global optimum. We conclude that our proposed algorithm framework has the simplest solution procedure and has potential efficiency advantages, which may reduce the complexity of enumerative schemes for medium or large-scale problems, while comparing with existing global algorithms such as the Kth-best algorithm (a two-level vertex-searching algorithm) and the single-level duality-based reformulation algorithm (a single-level vertex-searching algorithm).

DP-ADMM: ADMM-Based Distributed Learning With Differential Privacy

Alternating Direction Method of Multipliers (ADMM) is a widely used tool for machine learning in distributed settings, where a machine learning model is trained over distributed data sources through an interactive process of local computation and message passing. Such an iterative process could cause privacy concerns of data owners. The goal of this paper is to provide differential privacy for ADMM-based distributed machine learning. Prior approaches on differentially private ADMM exhibit low utility under high privacy guarantee and often assume the objective functions of the learning problems to be smooth and strongly convex. To address these concerns, we propose a novel differentially private ADMM-based distributed learning algorithm called DP-ADMM, which combines an approximate augmented Lagrangian function with time-varying Gaussian noise addition in the iterative process to achieve higher utility for general objective functions under the same differential privacy guarantee. We also apply the moments accountant method to bound the end-to-end privacy loss. The theoretical analysis shows that DP-ADMM can be applied to a wider class of distributed learning problems, is provably convergent, and offers an explicit utility-privacy tradeoff. To our knowledge, this is the first paper to provide explicit convergence and utility properties for differentially private ADMM-based distributed learning algorithms. The evaluation results demonstrate that our approach can achieve good convergence and model accuracy under high end-to-end differential privacy guarantee.

Distributed Continuous-Time and Discrete-Time Optimization With Nonuniform Unbounded Convex Constraint Sets and Nonuniform Stepsizes

This paper is devoted to distributed continuous-time and discrete-time optimization problems with nonuniform convex constraint sets and nonuniform stepsizes for general differentiable convex objective functions. The communication graphs are not required to be strongly connected at any time, the gradients of the local objective functions are not required to be bounded when their independent variables tend to infinity, and the constraint sets are not required to be bounded. For continuous-time multi-agent systems, a distributed continuous algorithm is first introduced where the stepsizes and the convex constraint sets are both nonuniform. It is shown that all agents reach a consensus while minimizing the team objective function even when the constraint sets are unbounded. After that, the obtained results are extended to discrete-time multi-agent systems and then the case where each agent remains in a corresponding convex constraint set is studied. To ensure all agents to remain in a bounded region, a switching mechanism is introduced in the algorithms. It is shown that the distributed optimization problems can be solved, even though the discretization of the algorithms might deviate the convergence of the agents from the minimum of the objective functions. Finally, numerical examples are included to show the obtained theoretical results.

Adjoint sensitivity analysis approach for the nonlinear Schrödinger equation.

This Letter proposes a novel adjoint sensitivity analysis approach for the nonlinear Schrödinger equation, which describes the light wave propagation in optical fiber communication systems. Using only one extra adjoint system simulation, all the sensitivities of a general objective function with respect to all fiber design parameters are estimated. We provide a full description of the solution to the derived adjoint problem. The accuracy and efficiency of our proposed algorithm are illustrated through optical fiber examples.

Search for Rational Parameters of Building Structures Based on Multi-Criteria Evolutionary Optimization

An approach to the design of building structures, in the general case, taking into account the four groups of optimality criteria at the main stages of their life cycle, is proposed. The first group is aimed at rational minimization of the structures cost at the stages of design, operation, reconstruction and disposal. The second group of criteria makes it possible to search for solutions with due regard for the minimization of the integral value of risks of social and economic consequences of possible emergencies. The third group is related to the assessment of the internal environment comfort of the premises, the last one - with the influence of the features of structures production and the functional process of the object during its operation on the surrounding biosphere. The solution of such a multi-criteria problem with a large number of variable parameters is proposed to obtain using high-performance genetic algorithms that use the Nash equilibrium principle for the "penalties" values of the generalized objective function, when the constraints on certain optimality criteria are not satisfied. The presented approach makes it possible to obtain design solutions rational from the perspective of a complex multi-criteria evaluation for bearing and enclosing structures.

The Short Path Algorithm Applied to a Toy Model

We numerically investigate the performance of the short path optimization algorithm on a toy problem, with the potential chosen to depend only on the total Hamming weight to allow simulation of larger systems. We consider classes of potentials with multiple minima which cause the adiabatic algorithm to experience difficulties with small gaps. The numerical investigation allows us to consider a broader range of parameters than was studied in previous rigorous work on the short path algorithm, and to show that the algorithm can continue to lead to speedups for more general objective functions than those considered before. We find in many cases a polynomial speedup over Grover search. We present a heuristic analytic treatment of choices of these parameters and of scaling of phase transitions in this model.

Nonconvex utility-based power allocation for cognitive radio MIMO system over fading channels

In this paper, we present a power control algorithm in cognitive radio multiple-input–multiple-output (MIMO) system over time-varying fading channels based on the utility-based framework. In particular, a more general objective function framework is proposed under a long- or short-term transmitting and interference power constraints for secondary users, which results the optimization problem is nonconvex. A stochastic optimization algorithm is proposed based on an improved quantum-behaved particle swarm optimization (IQPSO), which can solve the nonconvex optimization problem efficiently. The acceleration coefficients are adjusted with time-varying equations which can help the particles jump out the local optimum. Due to the characteristics of wave function of quantum Delta potential well model, IQPSO can get better optimal solutions in search space for the stochastic optimization problem. To show the efficiency of the proposed algorithm, the utilities of the MIMO CR system got by IQPSO are compared with other approaches in the literature in different case studies, which show that the proposed algorithm can solve the nonconvex problem efficiently and achieve significant throughput.

Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework

Hadamard’s method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem. The coupling is weak because the equations involved are solved consecutively, namely the steady state Navier–Stokes equations for the fluid domain, first, the convection diffusion equation for the whole domain, second, and the linear thermo-elasticity system in the solid domain, third. Shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions. An emphasis is given on the derivation of the adjoint interface condition dual to the one of equality of the normal stresses at the fluid solid interface. The arguments allowing to obtain this surprising condition are specifically detailed on a simplified scalar problem. Numerical test cases are presented using the level set mesh evolution framework of Allaire et al. (Appl Mech Eng 282:22–53, 2014). It is demonstrated how the implementation enables to treat a variety of shape optimization problems.

Optimal configuration of spaceborne bistatic SAR with GEO transmitter and LEO receiver

Spaceborne bistatic synthetic aperture radar (BiSAR) with a geosynchronous (GEO) transmitter and a low-earth-orbit (LEO) receiver can offer high signal-to-noise ratio, fine resolution and large surveillance area, which provides great potential for future Earth observation technology. Nevertheless, due to complex bistatic geometry and big differences of orbital characteristics, the configuration design of GEO-LEO BiSAR remains a challenge. To address the issue, mathematic models between the indicators related to the imaging performance and the configuration parameters are firstly derived based on the geometry analysis. Then, a general nonlinear multivariate objective function is presented for the desired imaging performance. Accordingly, the configuration design problem can be transformed to solve nonlinear multivariate optimisation equation. A novel optimal configuration design method based on the simulated annealing algorithm is presented. Finally, simulation results are provided for two typical observational missions to verify the effectiveness of the proposed method.

Multi-objective Inventory Model with Both Stock-Dependent Demand Rate and Holding Cost Rate Under Fuzzy Random Environment

In this paper, we investigated a multi-objective inventory model under both stock-dependent demand rate and holding cost rate with fuzzy random coefficients. Chance constrained fuzzy random multi-objective model and a traditional solution procedure based on an interactive fuzzy satisfying method are discussed. In addition, the technique of fuzzy random simulation is applied to deal with general fuzzy random objective functions and fuzzy random constraints which are usually difficult to converted into their crisp equivalents. The purposed of this study is to determine optimal order quantity and inventory level such that the total profit and wastage cost are maximized and minimize for the retailer respectively. Finally, illustrate example is given in order to show the application of the proposed model.

Engineering based robot optimization methodology

The paper deals with the engineering based methodology for optimal design of non-standard robotic architectures. Two layer algorithm is presented for the optimization of robot kinematic parameters. The first layer includes mathematical optimization of the simplified dynamic model of the robot resulting in a priori estimation of the kinematic parameters. A general objective function for robot joint force/torque minimization is taken into account. The second layer provides an iterative approach which makes possible to adjust the kinematic parameters according to unmodeled engineering requirements. The proposed optimization approach is illustrated on the example of 5 a DoF robot for industrial degreasing machine.

Mathematical model of consumer regulators management for alignment of electric load graphs of transformer substation 10/0.4 kV

The change of load of the transformer substation TS 10 / 0,4 kV in the mode of maximum and minimum load is measured. It is noted that the process is non-stationary at the daily in-terval. Hourly intervals of time are highlighted on the daily graph of electrical load, the study of the law of distribution is conducted, and the normal law of distribution is confirmed by Pearson's criterion. The stationarity test was performed on parametric tests, namely the verification of the Fisher's Fish-er's constant variance and the mathematical expectation of the Student's t-test and the correlation function.The values of numerical characteristics were obtained at the stationary areas and a probabilistic mathematical model of the TS 10 / 0.4 kV load was constructed. The method of equalization of the electric load graph of TS 10 / 0,4 kV is developed by optimal management of the power of consumers-regulators in urban electric networks, taking into account the features of the electric load graph of the main con-sumers of TS 10 / 0,4 kV. For this purpose, we set a target function, where the criterion of optimization is the total min-imum cost to cover the losses of electricity in the network caused by the irregularity of the electric load graph and the electricity consumed, a system of constraints caused by the load capacity of the transformers and the required energy consumption by the technology regulators has been compiled and boundary conditions determined by the installed capaci-ties of the consumers-regulators. In order to take into account the trade-off between the constituents of the objective func-tion, the weighted coefficients of the generalized objective function are chosen by the method of expert estimation. A nonlinear programming method was used to fulfill the opti-mization process, and the extremum of the function was found using the Newton method. The solution of the problem is implemented in MS Excel.

Distributed constrained optimization for multi-agent networks with nonsmooth objective functions

This paper investigates the distributed constrained optimization problem in multi-agent systems, where agents cooperatively minimize an objective function being the sum of each agent’s objective function while meeting equality and inequality constraints. By virtue of nonsmooth analysis and graph theory, a novel distributed continuous-time algorithm is proposed to solve such a kind of optimization problems. Different from existing continuous-time results relying on the differentiability or strict (strong) convexity of local objective functions, the proposed approach considers more general local objective functions which are only convex and not necessarily smooth. In addition, the proposed approach considers more general local constraints, not just box constraints considered in most existing studies. The optimality of the proposed algorithm is ensured under certain initial condition. Based on set-valued LaSalle invariance principle, the convergence of the proposed scheme is rigorously proved. Finally, simulation examples are applied to validate the effectiveness of the proposed algorithm.

On the Optimal Design of Lottery Contests

This paper develops a novel technique that allows us to characterize the optimal biased generalized lottery contest. In our baseline setting, we search for the optimal multiplicative biases for asymmetric Tullock contests — i.e., the weights placed on contestants' effort entries in the contest success function. Asymmetric Tullock contests, in general, have no closed-form solutions, which nullifies the usual implicit programming approach. We propose an alternative approach that allows us to circumvent this difficulty and characterize the optimum toward a wide array of objectives without solving for the equilibrium explicitly. The results of optimization exercises yield novel implications on the strategic nature of the contest game and its optimal design. In particular, we show that the conventional wisdom of leveling the playing field does not generally hold. Further, we relax restrictions on the functional forms, allow for enriched design space, and address more general objective functions. We show that our approach applies flexibly in a wide array of contexts and generates broad implications.

Restricted strong convexity implies weak submodularity

We connect high-dimensional subset selection and submodular maximization. Our results extend the work of Das and Kempe [In ICML (2011) 1057–1064] from the setting of linear regression to arbitrary objective functions. For greedy feature selection, this connection allows us to obtain strong multiplicative performance bounds on several methods without statistical modeling assumptions. We also derive recovery guarantees of this form under standard assumptions. Our work shows that greedy algorithms perform within a constant factor from the best possible subset-selection solution for a broad class of general objective functions. Our methods allow a direct control over the number of obtained features as opposed to regularization parameters that only implicitly control sparsity. Our proof technique uses the concept of weak submodularity initially defined by Das and Kempe. We draw a connection between convex analysis and submodular set function theory which may be of independent interest for other statistical learning applications that have combinatorial structure.

MODELING AND SOLVING THE TRAVELING SALESMAN PROBLEM WITH PRIORITY PRIZES

This paper addresses the Traveling Salesman Problem with Priority Prizes (TSPPP), an extension of the classical TSP in which the order of the node visits is taken into account in the objective function. A prize p ki is received by the traveling salesman when node i is visited in the k-th order of the route, while a travel cost c ij is incurred when the salesman travels from node i to node j . The aim of the TSPPP is to find the maximum profit n-node tour. The problem can be seen as a TSP variant with a more general objective function, aiming at solutions that in some way consider the quality of customer service and the delivery priorities and costs. A natural representation for the TSPPP is here grounded in the point of view of Koopmans and Beckmann approach, according to which the problem is seem as a special case of the quadratic assignment problem (QAP). Given the novelty of this TSP variant, we propose different mixed integer programming models to appropriately represent the TSPPP, some of them based on the QAP. Computational experiments are also presented when solving the MIP models with a well-known optimization software, as well as with a tabu search algorithm.

Parsimonious trajectory design of connected automated traffic

One challenging problem about connected automated vehicles is to optimize vehicle trajectories considering realistic constraints (e.g. vehicle kinematic limits and collision avoidance) and objectives (e.g., travel time, fuel consumption). With respect to communication cost and implementation difficulty, parsimonious trajectory planning has attracted continuous interests. In this paper, we first analyze the feasibility conditions for a general continuous-time trajectory planning problem and then propose an analytical solution method for two important boundary trajectory problems. We further propose a discrete-time model with a more general objective function and a certain sparsity requirement that helps parsimonious planned trajectories. This sparsity requirement is implemented with a l1 norm regulatory term appended to the objective function. Numerical examples are conducted on several representative applications and show that the proposed design strategy is effective.

A piecewise trajectory optimization model for connected automated vehicles: Exact optimization algorithm and queue propagation analysis

This paper formulates a simplified traffic smoothing model for guiding movements of connected automated vehicles on a general one-lane highway segment. Adapted from the shooting heuristic proposed by Zhou et al. (2017) and Ma et al. (2017), this model confines each vehicle’s trajectory as a piecewise quadratic function with no more than five pieces and lets all trajectories in the same platoon share identical acceleration and deceleration rates. Similar to the shooting heuristic, the proposed simplified model is able to control the overall smoothness of a platoon of connected automated vehicles and approximately optimize traffic performance in terms of fuel efficiency and driving comfort. While the shooting heuristic relies on numerical meta-heuristic algorithms that cannot ensure solution optimality, we discover a set of elegant theoretical properties for the general objective function and the associated constraints in the proposed simplified model, and consequentially propose an efficient analytical algorithm for solving this problem to the exact optimum. Interestingly, this exact algorithm has intuitive physical interpretations, i.e., stretching the transitional parts of the trajectories (i.e., parts with acceleration and deceleration adjustments) as far as they reach the upstream end of the investigated segment, and then balancing the acceleration and deceleration magnitudes as close as possible. This analytical exact model can be considered as a core module to a range of general trajectory optimization problems at various infrastructure settings. Numerical examples reveal that this exact algorithm has much more efficient computational performance and the same or better solution quality compared with the previously proposed shooting heuristic. These examples also illustrate how to apply this model to CAV control problems on signalized segments and at non-stop intersections. Further, we study a homogeneous special case of this model and analytically formulate the relationship between queue propagation and trajectory smoothing. One counter-intuitive finding is that trajectory smoothing may not always cause longer queue propagation but instead may mitigate queue propagation with appropriate settings. This theoretical finding has valuable implications to joint optimization of queuing management and traffic smoothing in complex transportation networks.

An efficient adaptive three-term extension of the Hestenes–Stiefel conjugate gradient method

A new three-term Hestenes–Stiefel-type conjugate gradient method is proposed in which the search direction can satisfy the sufficient descent condition as well as an adaptive conjugacy condition. It is notable that search directions of the method are dynamically adjusted between that of the Newton method and the 3HS+ method, accelerating the convergence or reducing the condition number of iteration matrix. Under mild conditions, we show that the proposed method converges globally for general objective functions. Numerical experiments indicate that the method is practically promising.

Unbounded parallel-batch scheduling with drop-line tasks

In this paper, we study unbounded parallel-batch scheduling with drop-line tasks to minimize a regular objective function, where by “drop-line tasks” we mean that the completion time of each task (job) is equal to the sum of the starting time of the batch containing the task and the processing time of the task. In the problems considered in this paper, we assume that the tasks have individual release dates and the general regular objective function to be minimized is either of the sum-form or of the max-form. We then study the computational complexity of these problems on an unbounded parallel-batch processor. We show that (i) the problems are binary NP-hard and are solvable in pseudo-polynomial times, and (ii) when the number of processing times or release dates is a constant, the problems are solvable in polynomial times. We also point out some consequences of approximation algorithms.