Additional Inequality Constraints Research Articles

Feasibility Verification and Upper Bound Computation in Global Minimization Using Approximate Active Index Sets

We propose a new upper bounding procedure for global minimization problems with continuous variables and possibly nonconvex inequality and equality constraints. Upper bounds are crucial for standard termination criteria of spatial branch-and-bound (SBB) algorithms to ensure that they can enclose globally minimal values sufficiently well. However, whereas for most lower bounding procedures from the literature, convergence on smaller boxes is established, this does not hold for several methods to compute upper bounds even though they often perform well in practice. In contrast, our emphasis is on the convergence. We present a new approach to verify the existence of feasible points on boxes, on which upper bounds can then be determined. To this end, we resort to existing convergent feasibility verification approaches for purely equality and box constrained problems. By considering carefully designed modifications of subproblems based on the approximation of active index sets, we enhance such methods to problems with additional inequality constraints. We prove that our new upper bounding procedure finds sufficiently good upper bounds so that termination of SBB algorithms is guaranteed after a finite number of iterations. Our theoretical findings are illustrated by computational results on a large number of standard test problems. These results show that compared with interval Newton methods from the literature, our proposed method is more successful in feasibility verification for both, a full SBB implementation (42 instead of 26 test problems) and exhaustive sequences of boxes around known feasible points (120 instead of 29 test problems). History: Accepted by Antonio Frangioni, Area Editor for Design & Analysis of Algorithms–Continuous. 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.2023.0162 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0162 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Sag-Free Initialization for Strand-Based Hybrid Hair Simulation

Lagrangian/Eulerian hybrid strand-based hair simulation techniques have quickly become a popular approach in VFX and real-time graphics applications. With Lagrangian hair dynamics, the inter-hair contacts are resolved in the Eulerian grid using the continuum method, i.e., the MPM scheme with the granular Drucker-Prager rheology, to avoid expensive collision detection and handling. This fuzzy collision handling makes the authoring process significantly easier. However, although current hair grooming tools provide a wide range of strand-based modeling tools for this simulation approach, the crucial sag-free initialization functionality remains often ignored. Thus, when the simulation starts, gravity would cause any artistic hairstyle to sag and deform into unintended and undesirable shapes. This paper proposes a novel four-stage sag-free initialization framework to solve stable quasistatic configurations for hybrid strand-based hair dynamic systems. These four stages are split into two global-local pairs. The first one ensures static equilibrium at every Eulerian grid node with additional inequality constraints to prevent stress from exiting the yielding surface. We then derive several associated closed-form solutions in the local stage to compute segment rest lengths, orientations, and particle deformation gradients in parallel. The second global-local step solves along each hair strand to ensure all the bend and twist constraints produce zero net torque on every hair segment, followed by a local step to adjust the rest Darboux vectors to a unit quaternion. We also introduce an essential modification for the Darboux vector to eliminate the ambiguity of the Cosserat rod rest pose in both initialization and simulation. We evaluate our method on a wide range of hairstyles, and our approach can only take a few seconds to minutes to get the rest quasistatic configurations for hundreds of hair strands. Our results show that our method successfully prevents sagging and has minimal impact on the hair motion during simulation.

A singular value decomposition based approach to handle ill-conditioning in optimization problems with applications to portfolio theory

We identify a source of numerical instability of quadratic programming problems that is hidden in its linear equality constraints. We propose a new theoretical approach to rewrite the original optimization problem in an equivalent reformulation using the singular value decomposition and substituting the ill-conditioned original matrix of the restrictions with a suitable optimally conditioned one. The proposed novel approach is showed, both empirically and theoretically, to solve ill-conditioning related numerical issues, not only when they depend on bad scaling and are relative easy to handle, but also when they result from almost collinearity or when numerically rank-deficient matrices are involved. Furthermore, our strategy looks very promising even when additional inequality constraints are considered in the optimization problem, as it occurs in several practical applications. In this framework, even if no closed form solution is available, we show, through empirical evidence, how the equivalent reformulation of the original problem greatly improves the performances of MatLab®’s quadratic programming solver and Gurobi®. The experimental validation is provided through numerical examples performed on real financial data in the portfolio optimization context.

On Generalization and Computation of Tukey's Depth: Part II

This paper studies how to generalize Tukey's depth to problems defined in a restricted space that may be curved or have boundaries, and to problems with a nondifferentiable objective. First, using a manifold approach, we propose a broad class of Riemannian
 depth for smooth problems defined on a Riemannian manifold, and showcase its applications in spherical data analysis, principal component analysis, and multivariate orthogonal regression. Moreover, for nonsmooth problems, we introduce additional slack variables and inequality constraints to define a novel slacked data depth, which can perform center-outward rankings of estimators arising from sparse learning and reduced rank regression. Real data examples illustrate the usefulness of some proposed data depths.

Minimum variance constrained estimator

This paper is concerned with the problem of state estimation for discrete-time linear systems in the presence of additional (equality or inequality) constraints on the state (or estimate). By use of the minimum variance duality, the estimation problem is converted into an optimal control problem. Two algorithmic solutions are described: the full information estimator (FIE) and the moving horizon estimator (MHE). The main result is to show that the proposed estimator is stable in the sense of an observer. The proposed algorithm is distinct from the standard algorithm for constrained state estimation based upon the use of the minimum energy duality. The two are compared numerically on the benchmark batch reactor process model.

Approximation of value function of differential game with minimal cost

The paper is concerned with the approximation of the value function of the zero-sum differential game with the minimal cost, i.e., the differential game with the payoff functional determined by the minimization of some quantity along the trajectory by the solutions of continuous-time stochastic games with the stopping governed by one player. Notice that the value function of the auxiliary continuous-time stochastic game is described by the Isaacs–Bellman equation with additional inequality constraints. The Isaacs–Bellman equation is a parabolic PDE for the case of stochastic differential game and it takes a form of system of ODEs for the case of continuous-time Markov game. The approximation developed in the paper is based on the concept of the stochastic guide first proposed by Krasovskii and Kotelnikova.

Constrained adaptive sampling for domain reduction in surrogate model generation: Applications to hydrogen production

AbstractWe propose a new approach for sampling domain reduction for efficient surrogate model generation. Currently, the standard procedure is to use box constraints for the independent variables when sampling the exact simulator. However, by including additional inequality constraints to account for interdependencies between these variables, we can drastically reduce the sampling domain and ensure consistency of unit operations. Moreover, we present a methodology for constructing surrogate models based on penalized regression and error‐maximization sampling. All these algorithms have been implemented as a free and open‐source software package. Through a case study on the water–gas shift reaction for hydrogen production, we show that sampling domain reduction reduces the required number of sampling points significantly and improves the accuracy of the surrogate model.

On a nonlocal Cahn–Hilliard model permitting sharp interfaces

A nonlocal Cahn–Hilliard model with a non-smooth potential of double-well obstacle type that promotes sharp interfaces in the solution is presented. To capture long-range interactions between particles, a nonlocal Ginzburg–Landau energy functional is defined which recovers the classical (local) model as the extent of nonlocal interactions vanish. In contrast to the local Cahn–Hilliard problem that always leads to diffuse interfaces, the proposed nonlocal model can lead to a strict separation into pure phases of the substance. Here, the lack of smoothness of the potential is essential to guarantee the aforementioned sharp-interface property. Mathematically, this introduces additional inequality constraints that, in a weak formulation, lead to a coupled system of variational inequalities which at each time instance can be restated as a constrained optimization problem. We prove the well-posedness and regularity of the semi-discrete and continuous in time weak solutions, and derive the conditions under which pure phases are admitted. Moreover, we develop discretizations of the problem based on finite element methods and implicit–explicit time-stepping methods that can be realized efficiently. Finally, we illustrate our theoretical findings through several numerical experiments in one and two spatial dimensions that highlight the differences in features of local and nonlocal solutions and also the sharp interface properties of the nonlocal model.

The Generalized Trust-Region Sub-Problem with Additional Linear Inequality Constraints—Two Convex Quadratic Relaxations and Strong Duality

In this paper, we study the problem of minimizing a general quadratic function subject to a quadratic inequality constraint with a fixed number of additional linear inequality constraints. Under a regularity condition, we first introduce two convex quadratic relaxations (CQRs), under two different conditions, that are minimizing a linear objective function over two convex quadratic constraints with additional linear inequality constraints. Then, we discuss cases where the CQRs return the optimal solution of the problem, revealing new conditions under which the underlying problem admits strong Lagrangian duality and enjoys exact semidefinite optimization relaxation. Finally, under the given sufficient conditions, we present necessary and sufficient conditions for global optimality of the problem and obtain a form of S-lemma for a system of two quadratic and a fixed number of linear inequalities.

On Local Non-Global Minimizers of Quadratic Optimization Problem with a Single Quadratic Constraint

In this paper, we consider the nonconvex quadratic optimization problem with a single quadratic constraint. First we give a theoretical characterization of the local non-global minimizers. Then we extend the recent characterization of the global minimizer via a generalized eigenvalue problem to the local non-global minimizers. Finally, we use these results to derive an efficient algorithm that finds the global minimizer of the problem with an additional linear inequality constraint.

A Two-Timescale Duplex Neurodynamic Approach to Mixed-Integer Optimization.

This article presents a two-timescale duplex neurodynamic approach to mixed-integer optimization, based on a biconvex optimization problem reformulation with additional bilinear equality or inequality constraints. The proposed approach employs two recurrent neural networks operating concurrently at two timescales. In addition, particle swarm optimization is used to update the initial neuronal states iteratively to escape from local minima toward better initial states. In spite of its minimal system complexity, the approach is proven to be almost surely convergent to optimal solutions. Its superior performance is substantiated via solving five benchmark problems.

Identification of Wiener models for dynamic and steady‐state performance with application to solid oxide fuel cell

AbstractThis work is concerned with identification of Wiener models (a linear dynamic part connected in series with a nonlinear dynamic one). A neural network with one hidden layer is used as the nonlinear block of the model, two network configurations are considered. For model identification three algorithms are described. In the first case model accuracy only in transient conditions is considered, only the dynamic data is used for model training. In the next two algorithms model accuracy in both transient and steady‐state conditions is considered, dynamic and steady‐state data sets are used. The steady‐state model errors are taken into account by an additional term in the minimized cost‐function or by additional inequality constraints. For comparison of discussed algorithms and model structures, identification of a Wiener model of a solid oxide fuel cell (SOFC) process is considered. It is shown that the best results are obtained by the algorithm 2 which minimizes at the same time both dynamic and steady‐state model errors, additional constraints used in the algorithm 3 are computationally quite demanding.

Determining a unique solution to underdetermined metabolic networks via a systematic path through the Most Accurate Fluxes

Abstract Metabolic Flux Analysis (MFA) and Flux Balance Analysis (FBA) often lead to underdetermined problems in the sense that there are more unknown fluxes in the metabolic network than the number of available equations which represent balanced metabolites and measured fluxes. Even the additional inequality constraints, e.g. flux positivity, and/or the use of an objective function in FBA do not allow obtaining a unique solution in many cases. This contribution aims at determining a simple, systematic and computationally efficient algorithm for obtaining a unique solution based on the iterative determination of the Most Accurate Fluxes (MAF). A measure of accuracy is introduced and the systematic algorithm is proposed and illustrated on a case study aiming at the determination of unique dynamic flux values within hybridoma cells. It is also shown that the MAF distribution is similar to the mean values obtained from a uniform sampling of admissible solutions.

Investigations on different Fischer‐Burmeister functions applied to the modelling of ferroelectrics

AbstractDifferent selected approaches of Fischer‐Burmeister (FB) nonlinear complementarity problem (NCP) functions are investigated towards the local convergence properties and the possible parameter influences. The Fischer‐Burmeister functions are applied to a micromechanically motivated material model for ferroelectric materials. The particular model employed is a laminate‐based modelling approach to simulate the single‐ and polycrystalline tetragonal ferroelectric material behaviour. In this model, the occurring variant fractions are characterised via laminate fractions with additional inequality constraints. To solve the complementarity conditions of these internal variables, the FB approach provides a very efficient scheme compared to common solution schemes as, e.g. active‐set strategies.

An improved differential evolution algorithm for optimization including linear equality constraints

A differential evolution algorithm (DE) is proposed to exactly satisfy the linear equality constraints present in a continuous optimization problem that may also include additional non-linear equality and inequality constraints. The proposed DE technique, denoted by DELEqC-II, is an extension of a previous method developed by the authors. In contrast to the previous approach, it uses both mutation and crossover strategies that maintain feasibility with respect to the linear equality constraints. Also, a procedure to correct numerical errors detected in the previous approach was incorporated in DELEqC-II. In the numerical experiments, scalable test-problems with linear equality constraints are used to analyze the performance of the new proposal.

Maximum-entropy sampling and the Boolean quadric polytope

We consider a bound for the maximum-entropy sampling problem (MESP) that is based on solving a max-det problem over a relaxation of the Boolean quadric polytope (BQP). This approach to MESP was first suggested by Christoph Helmberg over 15 years ago, but has apparently never been further elaborated or computationally investigated. We find that the use of a relaxation of BQP that imposes semidefiniteness and a small number of equality constraints gives excellent bounds on many benchmark instances. These bounds can be further tightened by imposing additional inequality constraints that are valid for the BQP. Duality information associated with the BQP-based bounds can be used to fix variables to 0/1 values, and also as the basis for the implementation of a “strong branching” strategy. A branch-and-bound algorithm using the BQP-based bounds solves some benchmark instances of MESP to optimality for the first time.

How to tighten a commonly used MINLP superstructure formulation for simultaneous heat exchanger network synthesis

MINLP superstructures for heat exchanger network synthesis allow the simultaneous optimization of utility heat loads, the number of heat exchanger units and their area requirements. The algorithms used to solve these nonlinear non-convex optimization problems solve MILP and NLP sub-problems iteratively to find optimal solutions. If these sub-problems are tightened, which means that the solution space is reduced but still includes all feasible integer solutions, the algorithms can potentially go through the solution space faster as branches can be excluded earlier. In this work, tightening measures for a commonly used MINLP stage-wise superstructure formulation are proposed and the impact of tighter variable bounds and additional inequality constraints is investigated using various case-studies taken from literature. It is shown that tighter formulations help the solver to find global optimal solutions and that the duality gap can be reduced significantly if the test cases could not be solved to global optimality.

Integrated online trajectory planning and optimization in distinctive topologies

This paper presents a novel integrated approach for efficient optimization based online trajectory planning of topologically distinctive mobile robot trajectories. Online trajectory optimization deforms an initial coarse path generated by a global planner by minimizing objectives such as path length, transition time or control effort. Kinodynamic motion properties of mobile robots and clearance from obstacles impose additional equality and inequality constraints on the trajectory optimization. Local planners account for efficiency by restricting the search space to locally optimal solutions only. However, the objective function is usually non-convex as the presence of obstacles generates multiple distinctive local optima.The proposed method maintains and simultaneously optimizes a subset of admissible candidate trajectories of distinctive topologies and thus seeking the overall best candidate among the set of alternative local solutions. Time-optimal trajectories for differential-drive and carlike robots are obtained efficiently by adopting the Timed-Elastic-Band approach for the underlying trajectory optimization problem. The investigation of various example scenarios and a comparative analysis with conventional local planners confirm the advantages of integrated exploration, maintenance and optimization of topologically distinctive trajectories.

Theory and application of p-regularized subproblems for p>2

The p-regularized subproblem (p-RS) is the key content of a regularization technique in computing a Newton-like step for unconstrained optimization. The idea is to incorporate a local quadratic approximation of the objective function with a weighted regularization term and then globally minimize it at each iteration. In this paper, we establish a complete theory of the p-RSs for general p>2 that covers previous known results on p=3 or p=4. The theory features necessary and sufficient optimality conditions for the global and also for the local non-global minimizers of (p-RS). It gives a closed-form expression for the global minimum set of (p-RS) and shows that (p-RS), p>2 can have at most one local non-global minimizer. Our theory indicates that (p-RS) have all properties that the trust region subproblems do. In application, (p-RS) can appear in natural formulation for optimization problems. We found two examples. One is to utilize the Tikhonov regularization to stabilize the least square solution for an over-determined linear system; and the other comes from numerical approximations to the generalized Ginzburg–Landau functionals. Moreover, when (p-RS) is appended with m additional linear inequality constraints, denoted by (p-), the problem becomes NP-hard. We show that the partition problem, the k-dispersion-sum problem and the quadratic assignment problem in combinatorial optimization can be equivalently formulated as special types of (p-RS) with p=4. In the end, we develop an algorithm for solving (p-RS).

Local nonglobal minima for solving large-scale extended trust-region subproblems

We study large-scale extended trust-region subproblems (eTRS) i.e., the minimization of a general quadratic function subject to a norm constraint, known as the trust-region subproblem (TRS) but with an additional linear inequality constraint. It is well known that strong duality holds for the TRS and that there are efficient algorithms for solving large-scale TRS problems. It is also known that there can exist at most one local non-global minimizer (LNGM) for TRS. We combine this with known characterizations for strong duality for eTRS and, in particular, connect this with the so-called hard case for TRS. We begin with a recent characterization of the minimum for the TRS via a generalized eigenvalue problem and extend this result to the LNGM. We then use this to derive an efficient algorithm that finds the global minimum for eTRS by solving at most three generalized eigenvalue problems.