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We present SLEQP, an open source C‐package consisting of an active‐set method capable of solving large‐scale nonlinear programming (NLP) problems based on successive linear programming and equality constrained quadratic programming techniques. The method includes a feasibility restoration phase and second‐order corrections (SOCs), achieving robust practical performance. It has also been adapted for the special cases of unconstrained optimization, box‐constrained problems, and nonlinear least squares problems and contains interfaces to both Python and MATLAB. To demonstrate the performance of the package, we perform a computational study based on the well‐known CUTest suite of NLP problems. [ABSTRACT FROM AUTHOR]

Column liquid chromatography plays a vital role in the downstream processing of biopharmaceuticals, where the goal is to extract and purify a target protein from a mixture. In this work, we examine a real‐world ion exchange chromatography process where electrostatic interactions between proteins and the chromatographic medium are the predominant forces being exploited for the separation of the participating species. More specifically, we successfully separate a recombinant dimeric fragment of the von Willebrand factor, used for the half‐life prolongation of certain proteins, from human serum albumin. Compared to the originally conducted experiments, the optimized experiment achieves a higher product purity in less time and with lower consumption of buffer salts, hence being favorable both in economical and ecological terms. [ABSTRACT FROM AUTHOR]

In recent years, building energy supply and distribution systems have become more complex, with an increasing number of energy generators, stores, flows, and possible combinations of operating modes. This poses challenges for supervisory control, especially when balancing the conflicting goals of maximizing comfort while minimizing costs and emissions to contribute to global climate protection objectives. Mixed‐integer nonlinear model predictive control is a promising approach for intelligent real‐time control that is able to properly address the specific characteristics and restrictions of building energy systems. We present a strategy that utilizes a decomposition approach, combining partial outer convexification with the Switch‐Cost Aware Rounding procedure to handle switching behavior and operating time constraints of building components in real‐time. The efficacy is demonstrated through practical applications in a single‐family home with a combined heat and power unit and in a multi‐family apartment complex with 18 residential units. Simulation studies show high correspondence to globally optimal solutions with significant cost savings potential of around 19%. [ABSTRACT FROM AUTHOR]

We propose an algorithm for solving bound-constrained mathematical programs with complementarity constraints on the variables. Each iteration of the algorithm involves solving a linear program with complementarity constraints in order to obtain an estimate of the active set. The algorithm enforces descent on the objective function to promote global convergence to B-stationary points. We provide a convergence analysis and preliminary numerical results on a range of test problems. We also study the effect of fixing the active constraints in a bound-constrained quadratic program that can be solved on each iteration in order to obtain fast convergence. [ABSTRACT FROM AUTHOR]

Approximation algorithms like sum-up rounding that allow to compute integer-valued approximations of the continuous controls in a weak ^* sense have attracted interest recently. They allow to approximate (optimal) feasible solutions of continuous relaxations of mixed-integer control problems (MIOCPs) with integer controls arbitrarily close. To this end, they use compactness properties of the underlying state equation, a feature that is tied to the infinite-dimensional vantage point. In this work, we consider a class of MIOCPs that are constrained by pointwise mixed state-control constraints. We show that a continuous relaxation that involves so-called vanishing constraints has beneficial properties for the described approximation methodology. Moreover, we complete recent work on a variant of the sum-up rounding algorithm for this problem class. In particular, we prove that the observed infeasibility of the produced integer-valued controls vanishes in an L^\infty -sense with respect to the considered relaxation. Moreover, we improve the bound on the control approximation error to a value that is asymptotically tight. [ABSTRACT FROM AUTHOR]

We consider the use of the polynomial chaos method to approximate optimal control problems with randomly varying uncertain parameters by deterministic surrogate problems. Our focus is on nonlinear problems that require a high expansion order to give meaningful statements about the uncertainty propagation. Their resulting size and complexity pose a computational challenge for traditional optimal control methods. Our contributions include an adaptive optimization strategy which refines the approximation quality separately for each state variable using suitable error estimates. The benefits are twofold: we obtain additional means for solution verification and reduce the computational effort for finding an approximate solution with increased precision, as is highlighted in a numerical case study with two nonlinear real-world problems. The algorithmic contribution is complemented by a convergence proof showing that the optimal control solutions approach the correct solution for increasing expansion orders. [ABSTRACT FROM AUTHOR]

Partial outer convexification has been used to derive relaxations of mixed-integer optimal control problems (MIOCPs) that are constrained by time-dependent differential equations. The family of sum-up rounding (SUR) algorithms provides a means to approximate feasible points of these relaxations, i.e., [0,1]-valued control trajectories, with {0,1}-valued points. The approximants computed by an SUR algorithm converge in a weak sense when the coarseness of the rounding grid of the SUR algorithm is driven to zero, which in turn induces norm convergence of the corresponding sequence of state vectors. We show that this approximation property can be transferred to MIOCPs with integer control variables distributed in more than one dimension when carrying out an appropriate grid refinement strategy. We deduce a norm convergence result for the state vector of elliptic PDE systems and provide computational results illustrating the applicability of the theoretical framework. [ABSTRACT FROM AUTHOR]

In this article, we present a unified framework for the numerical solution of optimal control problems (OCPs) constrained by ordinary differential equations with both implicit and explicit switches. We present the problem class and qualify different types of implicitly switched systems. This classification significantly affects opportunities for solving such problems numerically. By using techniques from generalized disjunctive programming, we transform the problem into a counterpart one wherein discontinuities no longer appear implicitly. Instead, the new problem contains discrete decision variables and vanishing constraints. Recent results from the field of mixed-integer optimal control theory enable us to omit integrality constraints on variables, and allow to solve a relaxed OCP. We use a ‘first discretize, then optimize’ approach to solve the problem numerically. A direct method based on adaptive collocation is used for the discretization. The resulting finite dimensional optimization problems are mathematical programs with vanishing constraints, and we discuss numerical techniques to solve sequences of this challenging problem class. To demonstrate the efficacy and merit of our proposed approach, we investigate three benchmark problems for hybrid dynamic systems. [ABSTRACT FROM AUTHOR]

Summary: Optimal control problems with parametric uncertainties frequently arise in control practice and can be addressed by means of a probabilistic robustification using the concept of chance constraints. In this article, we develop an efficient method based on the polynomial chaos expansion to compute nonlinear propagations of the reachable sets of all uncertain states and show how it can be used to approximate nonlinear and joint chance constraints. The strength of the obtained estimator in guaranteeing a satisfaction level is supported by providing an a priori error estimate with exponential convergence in case of sufficiently smooth solutions. The proposed approach is readily implemented in existing state‐of‐the‐art direct methods to optimal control and is evaluated for 2 real‐world nonlinear uncertain optimal control problems. The achieved level of robustness in terms of constraint satisfaction is verified by extensive Monte Carlo sampling. [ABSTRACT FROM AUTHOR]

Computational models of cognition provide an interface to connect advanced mathematical tools and methods to empirically supported theories of behavior in psychology, cognitive science, and neuroscience. In this article, we consider a computational model of instance-based learning, implemented in the ACT-R cognitive architecture. We propose an approach for obtaining mathematical reformulations of such cognitive models that improve their computational tractability. For the well-established Sugar Factory dynamic decision making task, we conduct a simulation study to analyze central model parameters. We show how mathematical optimization techniques can be applied to efficiently identify optimal parameter values with respect to different optimization goals. Beyond these methodological contributions, our analysis reveals the sensitivity of this particular task with respect to initial settings and yields new insights into how average human performance deviates from potential optimal performance. We conclude by discussing possible extensions of our approach as well as future steps towards applying more powerful derivative-based optimization methods. [ABSTRACT FROM AUTHOR]

A new method for solving sequences of quadratic programs (QPs) is presented. For each new QP in the sequence, the method utilizes hot-starts that employ information computed by an active-set QP solver during the solution of the first QP. This avoids the computation and factorization of the full constraint and Hessian matrices for all but the first problem in the sequence. The proposed algorithm can be seen as an extension of the iterative refinement procedure for linear systems to QP problems, coupled with the application of an accelerated linear solver method that employs hot-started QP solves as a preconditioner. Local convergence results are presented. The practical performance of the proposed method is demonstrated on a sequence of QPs arising in nonlinear model predictive control and during the solution of a set of randomly generated nonlinear optimization problems using sequential quadratic programming. In these experiments, the method proves to be fairly reliable, despite the lack of global convergence guarantees. The results also show a significant reduction in the computation time for large problems with dense constraint matrices, as well as in the number of matrix-vector products. [ABSTRACT FROM AUTHOR]

Mixed‐integer optimal control problems (MIOCPs) are often solved using a decomposition approach, consisting of solving a partial outer convexified relaxed problem and using a rounding scheme to obtain a discrete‐valued control. We show that even in the presence of Vanishing Constraints (VCs) a binary control with linearly bounded integrated control deviation exists, if the grid is equidistant. We note that this control can be computed efficiently and that the bound is tight. [ABSTRACT FROM AUTHOR]

Many optimal decision problems in scientific, engineering, and public sector applications involve both discrete decisions and nonlinear system dynamics that affect the quality of the final design or plan. These decision problems lead to mixed-integer nonlinear programming (MINLP) problems that combine the combinatorial difficulty of optimizing over discrete variable sets with the challenges of handling nonlinear functions. We review models and applications of MINLP, and survey the state of the art in methods for solving this challenging class of problems.Most solution methods for MINLP apply some form of tree search. We distinguish two broad classes of methods: single-tree and multitree methods. We discuss these two classes of methods first in the case where the underlying problem functions are convex. Classical single-tree methods include nonlinear branch-and-bound and branch-and-cut methods, while classical multitree methods include outer approximation and Benders decomposition. The most efficient class of methods for convex MINLP are hybrid methods that combine the strengths of both classes of classical techniques.Non-convex MINLPs pose additional challenges, because they contain non-convex functions in the objective function or the constraints; hence even when the integer variables are relaxed to be continuous, the feasible region is generally non-convex, resulting in many local minima. We discuss a range of approaches for tackling this challenging class of problems, including piecewise linear approximations, generic strategies for obtaining convex relaxations for non-convex functions, spatial branch-and-bound methods, and a small sample of techniques that exploit particular types of non-convex structures to obtain improved convex relaxations.We finish our survey with a brief discussion of three important aspects of MINLP. First, we review heuristic techniques that can obtain good feasible solution in situations where the search-tree has grown too large or we require real-time solutions. Second, we describe an emerging area of mixed-integer optimal control that adds systems of ordinary differential equations to MINLP. Third, we survey the state of the art in software for MINLP. [ABSTRACT FROM AUTHOR]

In this contribution, we address the efficient solution of optimal control problems of dynamic processes with many controls. Such problems arise, for example, from the outer convexification of integer control decisions. We treat this optimal control problem class using the direct multiple shooting method to discretize the optimal control problem. The resulting nonlinear problems are solved using sequential quadratic programming methods. We review the classical condensing algorithm that preprocesses the large but structured quadratic programs (QPs) to obtain small but dense ones. We show that this approach leaves room for improvement when applied in conjunction with outer convexification. To this end, we present a new complementary condensing algorithm for QPs with many controls. This algorithm is based on a hybrid null-space range-space approach to exploit the block structure of the QPs that is due to direct multiple shooting. An assessment of the theoretical run-time complexity reveals significant advantages of the proposed algorithm. We give a detailed account on the required number of floating point operations, depending on the process dimensions. Finally, we demonstrate the merit of the new complementary condensing approach by comparing the behaviour of both methods for a vehicle control problem in which the integer gear decision is convexified. [ABSTRACT FROM AUTHOR]

Abstract: We present advances in structure‐exploiting non‐intrusive polynomial chaos methods for the numerical solution of nonlinear optimal control problems with parametric uncertainties. In particular, we make use of the special structure induced by the spectral projection to reuse model derivatives and exploit sparsity information leading to a fast automatic sensitivity generation. This greatly reduces the computational effort of Newton‐type methods for the solution of the resulting high‐dimensional surrogate problem. The results are complemented by a complexity comparison and a numerical case study. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim) [ABSTRACT FROM AUTHOR]

We show that the class of unconstrained NLPs in abs‐normal form is a subclass of the class of MPECs and that the class of NLPs with general constraints in abs‐normal form is equivalent to the class of MPECs. Moreover, we compare constraint qualifications and stationarity concepts of these problem classes and observe close relations between them. [ABSTRACT FROM AUTHOR]

Mixed‐integer optimal control problems can be reformulated by means of partial outer convexification, which introduces binary‐valued switching functions for the different realizations of a discrete‐valued control variable. They can be relaxed naturally by allowing them to take values in [0, 1]. Sum‐Up Rounding (SUR) algorithms approximate feasible switching functions of the relaxation with binary ones. If the controls are distributed in one dimension, the approximants are known to converge in the weak∗ topology of L∞. We show that this still holds true for controls that are distributed in more than one dimension if an appropriate grid refinement strategy that is coupled with a deliberate ordering of the grid cells is chosen. This condition is satisfied by the iterates of space‐filling curves, e.g. the Hilbert curve. [ABSTRACT FROM AUTHOR]