Non-separable Objective Function Research Articles

Efficiency of Stochastic Coordinate Proximal Gradient Methods on Nonseparable Composite Optimization

This paper deals with composite optimization problems having the objective function formed as the sum of two terms; one has a Lipschitz continuous gradient along random subspaces and may be nonconvex, and the second term is simple and differentiable but possibly nonconvex and nonseparable. Under these settings, we design a stochastic coordinate proximal gradient method that takes into account the nonseparable composite form of the objective function. This algorithm achieves scalability by constructing at each iteration a local approximation model of the whole nonseparable objective function along a random subspace with user-determined dimension. We outline efficient techniques for selecting the random subspace, yielding an implementation that has low cost per iteration, also achieving fast convergence rates. We present a probabilistic worst case complexity analysis for our stochastic coordinate proximal gradient method in convex and nonconvex settings; in particular, we prove high-probability bounds on the number of iterations before a given optimality is achieved. Extensive numerical results also confirm the efficiency of our algorithm. Funding: This work was supported by Norway Grants 2014-2021 [Grant ELO-Hyp 24/2020]; Unitatea Executiva pentru Finantarea Invatamantului Superior, a Cercetarii, Dezvoltarii si Inovarii [Grants PN-III-P4-PCE-2021-0720, L2O-MOC, nr 70/2022]; and the ITN-ETN project TraDE-OPT funded by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Skłodowska-Curie grant agreement [Grant 861137].

Convex and Nonconvex Risk-Based Linear Regression at Scale

The value at risk (VaR) and the conditional value at risk (CVaR) are two popular risk measures to hedge against the uncertainty of data. In this paper, we provide a computational toolbox for solving high-dimensional sparse linear regression problems under either VaR or CVaR measures, the former being nonconvex and the latter convex. Unlike the empirical risk (neutral) minimization models in which the overall losses are decomposable across data, the aforementioned risk-sensitive models have nonseparable objective functions so that typical first order algorithms are not easy to scale. We address this scaling issue by adopting a semismooth Newton-based proximal augmented Lagrangian method of the convex CVaR linear regression problem. The matrix structures of the Newton systems are carefully explored to reduce the computational cost per iteration. The method is further embedded in a majorization–minimization algorithm as a subroutine to tackle the nonconvex VaR-based regression problem. We also discuss an adaptive sieving strategy to iteratively guess and adjust the effective problem dimension, which is particularly useful when a solution path associated with a sequence of tuning parameters is needed. Extensive numerical experiments on both synthetic and real data demonstrate the effectiveness of our proposed methods. In particular, they are about 53 times faster than the commercial package Gurobi for the CVaR-based sparse linear regression with 4,265,669 features and 16,087 observations. History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. Funding: This work was supported in part by the NSF, the Division of Computing and Communication Foundations [Grant 2153352], the National Natural Science Foundation of China [Grant 12271187], and the Hong Kong Research Grant Council [Grant 15304019]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.1282 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0012 ) at ( http://dx.doi.org/10.5281/zenodo.7483279 ).

Optimal Distributed Control of AC Microgrids With Coordinated Voltage Regulation and Reactive Power Sharing

In this paper, we propose an optimal distributed voltage control for grid-forming (GFM) inverters in islanded AC microgrids. An optimization problem is formulated where the distributed generator (DG) output voltage is considered as the control variable with technical constraints on voltage and reactive power output capacity and an objective function that makes a trade-off between voltage regulation and reactive power sharing. A distributed primal-dual gradient based algorithm is developed to solve the formulated optimization problem to address the challenges due to non-separable objective function, unavailable global average voltage, and globally coupled reactive power constraints. The effectiveness of the proposed optimal distributed control is validated through simulations on the 4-DG test microgrid and the modified IEEE 34-bus distribution test system, and the advantages of the proposed control over existing controls are demonstrated.

Dynamics analysis of a novel hybrid deep clustering for unsupervised learning by reinforcement of multi-agent to energy saving in intelligent buildings

The heating, ventilating and air conditioning (HVAC) systems energy demand can be reduced by manipulating indoor conditions within the comfort range, which relates to control performance and, simultaneously, achieves peak load shifting toward off-peak hours. Reinforcement learning (RL) is considered a promising technique to solve this problem without an analytical approach, but it has been unable to overcome the awkwardness of an extremely large action space in the real world; it would be quite hard to converge to a set point. The core of the problem with RL is its state space and action space of multi-agent action for building and HVAC systems that have an extremely large amount of training data sets. This makes it difficult to create weights layers accurately of the black-box model. Despite the efforts of past works carried out on deep RL, there are still drawback issues that have not been dealt with as part of the basic elements of large action space and the large-scale nonlinearity due to high thermal inertia. The hybrid deep clustering of multi-agent reinforcement learning (HDCMARL) has the ability to overcome these challenges since the hybrid deep clustering approach has a higher capacity for learning the representation of large space and massive data. The framework of RL agents is a greedy iterative trained and organized as a hybrid layer clustering structure to be able to deal with a non-convex, non-linear and non-separable objective function. The parameters of the hybrid layer are optimized by using the Quasi-Newton (QN) algorithm for fast response signals of agents. That is to say, the main motivation is that the state and action space of multi-agent actions for building HVAC controls are exploding, and the proposed method can overcome this challenge and achieve 32% better performance in energy savings and 21% better performance in thermal comfort than PID.

Multi-Agent Deep Reinforcement Learning for HVAC Control in Commercial Buildings

In commercial buildings, about 40%-50% of the total electricity consumption is attributed to Heating, Ventilation, and Air Conditioning (HVAC) systems, which places an economic burden on building operators. In this paper, we intend to minimize the energy cost of an HVAC system in a multi-zone commercial building with the consideration of random zone occupancy, thermal comfort, and indoor air quality comfort. Due to the existence of unknown thermal dynamics models, parameter uncertainties (e.g., outdoor temperature, electricity price, and number of occupants), spatially and temporally coupled constraints associated with indoor temperature and CO2 concentration, a large discrete solution space, and a non-convex and non-separable objective function, it is very challenging to achieve the above aim. To this end, the above energy cost minimization problem is reformulated as a Markov game. Then, an HVAC control algorithm is proposed to solve the Markov game based on multi-agent deep reinforcement learning with attention mechanism. The proposed algorithm does not require any prior knowledge of uncertain parameters and can operate without knowing building thermal dynamics models. Simulation results based on real-world traces show the effectiveness, robustness and scalability of the proposed algorithm.

Extensions of the Dynamic Programming Framework: Battery Scheduling, Demand Charges, and Renewable Integration

We consider a general class of dynamic programming (DP) problems with nonseparable objective functions. We show that for any problem in this class, there exists an augmented-state DP problem that satisfies the principle of optimality and the solutions to which yield solutions to the original problem. Furthermore, we identify a subclass of DP problems with naturally forward separable objective functions for which this state-augmentation scheme is tractable. We extend this framework to stochastic DP problems, proposing a suitable definition of the principle of optimality. We then apply the resulting algorithms to the problem of optimal battery scheduling with demand charges using a data-based stochastic model for electricity usage and solar generation by the consumer.

Decompositions of Semidefinite Matrices and the Perspective Reformulation of Nonseparable Quadratic Programs

We study the problem of decomposing the Hessian matrix of a mixed integer convex quadratic program (MICQP) into the sum of positive semidefinite 2 × 2 matrices. Solving this problem enables the use of perspective reformulation techniques for obtaining strong lower bounds for MICQPs with semicontinuous variables but a nonseparable objective function. An explicit formula is derived for constructing 2 × 2 decompositions when the underlying matrix is weakly scaled diagonally dominant, and necessary and sufficient conditions are given for the decomposition to be unique. For matrices lying outside this class, two exact semidefinite programming approaches and an efficient heuristic are developed for finding approximate decompositions. We present preliminary results on the bound strength of a 2 × 2 perspective reformulation for the portfolio optimization problem, showing that, for some classes of instances, the use of 2 × 2 matrices can significantly improve the quality of the bound with respect to the best previously known approach, although at a possibly high computational cost.

Decentralized Charging Control of Electric Vehicles in Residential Distribution Networks

Electric vehicle (EV) charging can negatively impact electric distribution networks by exceeding equipment thermal ratings and causing voltages to drop below standard ranges. In this paper, we develop a decentralized EV charging control scheme to achieve "valley-filling" (i.e., flattening demand profile during overnight charging), meanwhile meeting heterogeneous individual charging requirements and satisfying distribution network constraints. The formulated problem is an optimization problem with a non-separable objective function and strongly coupled inequality constraints. We propose a novel shrunken primal-dual subgradient (SPDS) algorithm to support the decentralized control scheme, derive conditions guaranteeing its convergence, and verify its efficacy and convergence with a representative distribution network model.

Enhanced Particle Swarm Optimization Algorithm: Efficient Training of ReaxFF Reactive Force Fields.

Particle swarm optimization (PSO) is a powerful metaheuristic population-based global optimization algorithm. However, when it is applied to nonseparable objective functions, its performance on multimodal landscapes is significantly degraded. Here we show that a significant improvement in the search quality and efficiency on multimodal functions can be achieved by enhancing the basic rotation-invariant PSO algorithm with isotropic Gaussian mutation operators. The new algorithm demonstrates superior performance across several nonlinear, multimodal benchmark functions compared with the rotation-invariant PSO algorithm and the well-established simulated annealing and sequential one-parameter parabolic interpolation methods. A search for the optimal set of parameters for the dispersion interaction model in the ReaxFF- lg reactive force field was carried out with respect to accurate DFT-TS calculations. The resulting optimized force field accurately describes the equations of state of several high-energy molecular crystals where such interactions are of crucial importance. The improved algorithm also presents better performance compared to a genetic algorithm optimization method in the optimization of the parameters of a ReaxFF- lg correction model. The computational framework is implemented in a stand-alone C++ code that allows the straightforward development of ReaxFF reactive force fields.

Optimizing the half-product and related quadratic Boolean functions: approximation and scheduling applications

This paper reviews the problems of Boolean non-linear programming related to the half-product problem. All problems under consideration have a similar quadratic non-separable objective function. For these problems, we focus on the development of fully polynomial-time approximation schemes, especially of those with strongly polynomial time, and on their applications to various scheduling problems.

The symmetric quadratic knapsack problem: approximation and scheduling applications

This paper reviews two problems of Boolean non-linear programming: the Symmetric Quadratic Knapsack Problem and the Half-Product Problem. The problems are related since they have a similar quadratic non-separable objective function. For these problems, we focus on the development of fully polynomial-time approximation schemes, especially of those with strongly polynomial time, and on their applications to various scheduling problems.

A Stackelberg equilibrium for a missile procurement problem

This paper deals with a procurement problem of missiles involving the efficient assignment of the missiles to some targets. Within a fixed amount of budget, a leader purchases several types of missiles, by which he aims to damage as much value as possible a follower hides into some facilities later. The effectiveness of the missile depends on the type of missile and facility. A payoff of the game is the expected amount of destroyed value. The problem is generalized as a two-person zero-sum game of distributing discrete resources with a leader and a follower. Our problem is to derive a Stackelberg equilibrium for the game. This type of game has an abundance of applications. The problem is first formulated into an integer programming problem with a non-separable objective function of variables and it is further equivalently transformed into a maximin integer knapsack problem. We propose three exacts methods and an approximation method for an optimal solution.

Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

In this paper we present a method for converting general nonlinear programming (NLP) problems into separable programming (SP) problems by using feedforward neural networks (FNNs). The basic idea behind the method is to use two useful features of FNNs: their ability to approximate arbitrary continuous nonlinear functions with a desired degree of accuracy and their ability to express nonlinear functions in terms of parameterized compositions of functions of single variables. According to these two features, any nonseparable objective functions and/or constraints in NLP problems can be approximately expressed as separable functions with FNNs. Therefore, any NLP problems can be converted into SP problems. The proposed method has three prominent features. (a) It is more general than existing transformation techniques; (b) it can be used to formulate optimization problems as SP problems even when their precise analytic objective function and/or constraints are unknown; (c) the SP problems obtained by the proposed method may highly facilitate the selection of grid points for piecewise linear approximation of nonlinear functions. We analyze the computational complexity of the proposed method and compare it with an existing transformation approach. We also present several examples to demonstrate the method and the performance of the simplex method with the restricted basis entry rule for solving SP problems.

Converting general nonlinear programming problems into separable programming problems with feedforward neural networks

In this paper we present a method for converting general nonlinear programming (NLP) problems into separable programming (SP) problems by using feedforward neural networks (FNNs). The basic idea behind the method is to use two useful features of FNNs: their ability to approximate arbitrary continuous nonlinear functions with a desired degree of accuracy and their ability to express nonlinear functions in terms of parameterized compositions of functions of single variables. According to these two features, any nonseparable objective functions and/or constraints in NLP problems can be approximately expressed as separable functions with FNNs. Therefore, any NLP problems can be converted into SP problems. The proposed method has three prominent features. (a) It is more general than existing transformation techniques; (b) it can be used to formulate optimization problems as SP problems even when their precise analytic objective function and/or constraints are unknown; (c) the SP problems obtained by the proposed method may highly facilitate the selection of grid points for piecewise linear approximation of nonlinear functions. We analyze the computational complexity of the proposed method and compare it with an existing transformation approach. We also present several examples to demonstrate the method and the performance of the simplex method with the restricted basis entry rule for solving SP problems.

Active-reactive coupling in optimal reactive dispatch: a solution via Karush-Kuhn-Tucker optimality conditions

Decoupled model based algorithms have been widely used to solve the optimal reactive dispatch problem. However, in the presence of nonnegligible active-reactive coupling, convergence and solution reliability problems may appear. This paper presents the implementation of an algorithm based on the estimation of active power related Lagrange multipliers to solve the optimal reactive dispatch. The estimation process is based on the Karush-Kuhn-Tucker optimality conditions. Adequate treatment of nonseparable objective functions and numerical results carried out on test-cases (17 buses) and real-life networks (329 and 447 buses) are reported. >

Experiments with dynamic programming algorithms for nonseparable problems

In this paper we report on numerical experiments with dynamic programming algorithms designed for solving problems with nonseparable objective functions of the C-programming type. The experiments were conducted with the view to evaluate the efficiency of C-programming's exclusionary rules when they are incorporated in the conventional parametric Lagrangian search method. Two classical nonseparable problems are considered: (1) the variance minimization problem and (2) the nonlinear fractional knapsack problem. The results indicate that the incorporation of C-programming's exclusionary rules in the Lagrangian procedure significantly improves the efficiency of the line search procedure.

On a discrete nonlinear and nonseparable knapsack problem

In this paper, a discrete knapsack problem with a convex but nonseparable objective function is considered. Relaxing the discrete requirements (and assuming nonnegativity), we show how the resulting problem can be solved by an efficient algorithm which solves a series of quadratic knapsack problems in O( n) time. This algorithm forms the basis for a branch-and-bound algorithm, which is subsequently used to test the efficacy of a proposed heuristic. Computational results are presented.

Continuum-type optimality criteria methods for large finite element systems with a displacement constraint. Part II

Continuum-type optimality criteria for the iterative optimization of “large” finite element systems (i.e. systems with over ten thousand finite elements) are discussed. By investigating optimization problems with up to one million elements and one million variables, it is shown that for a single displacement constraint the proposed method results in a rapid and almost uniform convergence, the rate of which, even for relatively ill-conditioned problems, does not depend significantly on the number of elements. Additional refinements, including upper and lower limits on the cross-sectional dimensions, segmentation, allowance for selfweight and the cost of supports, non-linear and non-separable objective functions, inclusion of shear deformations, built-up cross-sections as well as additional stress constraints and two-dimensional (plane stress) problems, will be considered in Part II of this contribution. The current development constitutes an extension and generalization of pioneering work by Berke, Khot, Venkayya and their associates, whose methods are also reviewed herein. In addition to an elementary truss example and a more advanced beam example, some simple layout optimization problems are considered in this Part. A special feature of the paper is that all numerical results presented are confirmed by closed form analytical solutions.

Piecewise-Linear Approximation Methods for Nonseparable Convex Optimization

An algorithm is described for the solution of nonseparable convex optimization problems. This method utilizes iterative piecewise-linear approximation of the nonseparable objective function, but requires function values only along a translated set of axes, thereby avoiding the curse of dimensionality commonly associated with grid methods for multi-dimensional problems. A global convergence proof is given under the assumptions that the objective function is Lipschitz continuous and differentiable and that the feasible set is convex and compact. The method is well-suited to linearly constrained large-scale optimization, since the direction-finding problems reduce to linear programs of manageable size. It is particularly appropriate for nonlinear networks, since it preserves the network structure of the constraints. In addition, because the resulting objective function approximation is separable, this approach permits for certain problem classes a decomposition that may be exploited for parallel computation. Some numerical results on the CRYSTAL multicomputer are presented to illustrate this decomposition feature.

Duality for programs involving non-separable objective functions

Mathematical programs involving non-separable objective functions arise in a variety of situations. In this paper we construct a duality theory for such functions using the notions of conjugate functions. Depending on the structure of the constraint set, different duals are possible. A particular case is analysed in detail, leading to an analytic solution. A numerical example is given.