Coradiant sets and $$\varepsilon $$-efficiency in multiobjective optimization

In multi-parametric programming an optimization problem is solved as a function of certain parameters, where the parameters are commonly considered to be bounded and continuous. In this paper, we use the case of strictly convex multi-parametric quadratic programming (mp-QP) problems with affine constraints to investigate problems where these conditions are not met. Based on the combinatorial solution approach for mp-QP problems featuring bounded and continuous parameters, we show that (i) for unbounded parameters, it is possible to obtain the multi-parametric solution if there exists one realization of the parameters for which the optimization problem can be solved and (ii) for binary parameters, we present the equivalent mixed-integer formulations for the application of the combinatorial algorithm. These advances are combined into a new, generalized version of the combinatorial algorithm for mp-QP problems, which enables the solution of problems featuring both unbounded and binary parameters. This novel approach is applied to mixed-integer bilevel optimization problems and the parametric solution of the dual of a convex problem.     

Piecewise polyhedral formulations for a multilinear term

In this paper, we present a mixed-integer linear programming (MILP) formulation of a piecewise, polyhedral relaxation (PPR) of a multilinear term using its convex-hull representation. Based on the PPR’s solution, we also present a MILP formulation whose solutions are feasible for nonconvex, multilinear equations. We then present computational results showing the effectiveness of proposed formulations on standard benchmark nonlinear programs (NLPs) with multilinear terms and compare with a traditional formulation that is built using recursive bilinear groupings of multilinear terms.     

Some results on the strength of relaxations of multilinear functions

We study approaches for obtaining convex relaxations of global optimization problems containing multilinear functions. Specifically, we compare the concave and convex envelopes of these functions with the relaxations that are obtained with a standard relaxation approach, due to McCormick. The standard approach reformulates the problem to contain only bilinear terms and then relaxes each term independently. We show that for a multilinear function having a single product term, this approach yields the convex and concave envelopes if the bounds on all variables are symmetric around zero. We then review and extend some results on conditions when the concave envelope of a multilinear function can be written as a sum of concave envelopes of its individual terms. Finally, for bilinear functions we prove that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is always within a constant of the difference between the concave and convex envelopes. These results, along with numerical examples we provide, give insight into how to construct strong relaxations of multilinear functions.     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Nonsmooth bifurcations of equilibria in planar continuous systems

In this paper we present a procedure to find all limit sets near bifurcating equilibria in a class of hybrid systems described by continuous, piecewise smooth differential equations. For this purpose, the dynamics near the bifurcating equilibrium is locally approximated as a piecewise affine systems defined on a conic partition of the plane. To guarantee that all limit sets are identified, conditions for the existence or absence of limit cycles are presented. Combining these results with the study of return maps, a procedure is presented for a local bifurcation analysis of bifurcating equilibria in continuous, piecewise smooth systems. With this procedure, all limit sets that are created or destroyed by the bifurcation are identified in a computationally feasible manner.     

Global optimization of multi-parametric MILP problems

In this paper, we present a novel global optimisation approach for the general solution of multi-parametric mixed integer linear programs (mp-MILPs). We describe an optimisation procedure which iterates between a (master) mixed integer nonlinear program and a (slave) multi-parametric program. Moreover, we explain how to overcome the presence of bilinearities, responsible for the non-convexity of the multi-parametric program, in two classes of mp-MILPs, with (i) varying parameters in the objective function and (ii) simultaneous presence of varying parameters in the objective function and the right-hand side of the constraints. Examples are provided to illustrate the solution steps.     

Piecewise affine functions and polyhedral sets ∗

In this paper we present a number of characterizations of piecewise affine and piecewise linear functions defined on finite dimesional normed vector spaces. In particular we prove that a real-valued function is piecewise affine [resp. piecewise linear] if both its epigraph and its hypograph are (nonconvex) polyhedral sets[resp..Polyhedral cones]. Also,We show that the collection of all piecewise affine[resp.piecewise linear] functions. Furthermore, we prove that a function is piecewise affine[resp.piecewise linear] if it can be represented as a difference of two convex [resp.,sublinear] polyhedral fucntions.     

Bandwidth packing problem with queueing delays: modelling and exact solution approach

In this paper we develop a general but smooth global optimization strategy for nonlinear multilevel programming problems with polyhedral constraints. At each decision level successive convex relaxations are applied over the non-convex terms in combination with a multi-parametric programming approach. The proposed algorithm reaches the approximate global optimum in a finite number of steps through the successive subdivision of the optimization variables that contribute to the non-convexity of the problem and partitioning of the parameter space. The method is implemented and tested for a variety of bilevel, trilevel and fifth level problems which have non-convexity formulation at their inner levels.     

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of \(\mathbb R^m\) were studied in Aliprantis et al. (Macroecon Dyn 10(1):77–99, 2006), Aliprantis et al. (J Econometrics 136(2):431–456, 2007), Aliprantis and Tourky (Cones and duality, 2007), Ovchinnikov (Beitr\(\ddot{\mathrm{a}}\)ge Algebra Geom 43:297–302, 2002). In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in \(C(\mathbb R^m)\), while piecewise affine functions are sequentially order dense in \(C(\mathbb R^m)\). This paper is partially based on Adeeb (Locally piece-wise affine functions, 2014)     

Constructive Solution of Inverse Parametric Linear/Quadratic Programming Problems

Parametric convex programming has received a lot of attention, since it has many applications in chemical engineering, control engineering, signal processing, etc. Further, inverse optimality plays an important role in many contexts, e.g., image processing, motion planning. This paper introduces a constructive solution of the inverse optimality problem for the class of continuous piecewise affine functions. The main idea is based on the convex lifting concept. Accordingly, an algorithm to construct convex liftings of a given convexly liftable partition will be put forward. Following this idea, an important result will be presented in this article: Any continuous piecewise affine function defined over a polytopic partition is the solution of a parametric linear/quadratic programming problem. Regarding linear optimal control, it will be shown that any continuous piecewise affine control law can be obtained via a linear optimal control problem with the control horizon at most equal to 2 prediction steps.     

Stabilization of affine control systems with periodic coefficients

We obtain new sufficient conditions for the uniform global asymptotic stabilization of the zero solution of an affine control system with periodic coefficients and some corollaries for bilinear control systems.     

Suboptimal Explicit Receding Horizon Control via Approximate Multiparametric Quadratic Programming

Algorithms for solving multiparametric quadratic programming (MPQP) were recently proposed in Refs. 1–2 for computing explicit receding horizon control (RHC) laws for linear systems subject to linear constraints on input and state variables. The reason for this interest is that the solution to MPQP is a piecewise affine function of the state vector and thus it is easily implementable online. The main drawback of solving MPQP exactly is that, whenever the number of linear constraints involved in the optimization problem increases, the number of polyhedral cells in the piecewise affine partition of the parameter space may increase exponentially. In this paper, we address the problem of finding approximate solutions to MPQP, where the degree of approximation is arbitrary and allows to tradeoff between optimality and a smaller number of cells in the piecewise affine solution. We provide analytic formulas for bounding the errors on the optimal value and the optimizer, and for guaranteeing that the resulting suboptimal RHC law provides closed-loop stability and constraint fulfillment.     

Deterministic solution approach for some classes of nonlinear multilevel programs with multiple followers

In this paper we investigate multilevel programming problems with multiple followers in each hierarchical decision level. It is known that such type of problems are highly non-convex and hard to solve. A solution algorithm have been proposed by reformulating the given multilevel program with multiple followers at each level that share common resources into its equivalent multilevel program having single follower at each decision level. Even though, the reformulated multilevel optimization problem may contain non-convex terms at the objective functions at each level of the decision hierarchy, we applied multi-parametric branch-and-bound algorithm to solve the resulting problem that has polyhedral constraints. The solution procedure is implemented and tested for a variety of illustrative examples.     

Stabilization of stationary affine control systems with discrete time

We obtain new sufficient conditions for the local and global asymptotic stabilization of the zero solution of a nonlinear affine control system with discrete time and with constant coefficients by a continuous state feedback. We assume that the zero solution of the free system is Lyapunov stable. For systems with linear drift, we construct a bounded control in the problem of global asymptotic state and output stabilization. Corollaries for bilinear systems are obtained.     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.     

Stabilization of irregular systems

We consider the stabilization problem for the zero equilibrium of bilinear and affine systems in canonical form. We obtain necessary and sufficient conditions for the stabilizability of second-order bilinear and affine systems in canonical form and generalize these conditions to the case of simultaneous stabilization of a family of bilinear systems and to nth-order bilinear systems.     

Existence of piecewise affine Lyapunov functions in two dimensions

In Marinosson (2002) [10], a method to compute Lyapunov functions for systems with asymptotically stable equilibria was presented. The method uses finite differences on finite elements to generate a linear programming problem for the system in question, of which every feasible solution parameterises a piecewise affine Lyapunov function. In Hafstein (2004) [2] it was proved that the method always succeeds in generating a Lyapunov function for systems with an exponentially stable equilibrium. However, the proof could not guarantee that the generated function has negative orbital derivative locally in a small neighbourhood of the equilibrium. In this article we give an example of a system, where no piecewise affine Lyapunov function with the proposed triangulation scheme exists. This failure is due to the triangulation of the method being too coarse at the equilibrium, and we suggest a fan-like triangulation around the equilibrium. We show that for any two-dimensional system with an exponentially stable equilibrium there is a local triangulation scheme such that the system possesses a piecewise affine Lyapunov function. Hence, the method might eventually be equipped with an improved triangulation scheme that does not have deficits locally at the equilibrium.     

New formulation and relaxation to solve a concave-cost network flow problem

In this paper we study a minimum cost, multicommodity network flow problem in which the total cost is piecewise linear, concave of the total flow along the arcs. Specifically, the problem can be defined as follows. Given a directed network, a set of pairs of communicating nodes and a set of available capacity ranges and their corresponding variable and fixed cost components for each arc, the problem is to select for each arc a range and identify a path for each commodity between its source and destination nodes so as to minimize the total costs. We also extend the problem to the case of piecewise nonlinear, concave cost function. New mathematical programming formulations of the problems are presented. Efficient solution procedures based on Lagrangean relaxations of the problems are developed. Extensive computational results across a variety of networks are reported. These results indicate that the solution procedures are effective for a wide range of traffic loads and different cost structures. They also show that this work represents an improvement over previous work made by other authors. This improvement is the result of the introduction of the new formulations of the problems and their relaxations.