Convex Feasible Region Research Articles

Generalization Bounds in the Predict-Then-Optimize Framework

The predict-then-optimize framework is fundamental in many practical settings: predict the unknown parameters of an optimization problem and then solve the problem using the predicted values of the parameters. A natural loss function in this environment is to consider the cost of the decisions induced by the predicted parameters in contrast to the prediction error of the parameters. This loss function is referred to as the smart predict-then-optimize (SPO) loss. In this work, we seek to provide bounds on how well the performance of a prediction model fit on training data generalizes out of sample in the context of the SPO loss. Because the SPO loss is nonconvex and non-Lipschitz, standard results for deriving generalization bounds do not apply. We first derive bounds based on the Natarajan dimension that, in the case of a polyhedral feasible region, scale at most logarithmically in the number of extreme points but, in the case of a general convex feasible region, have linear dependence on the decision dimension. By exploiting the structure of the SPO loss function and a key property of the feasible region, which we denote as the strength property, we can dramatically improve the dependence on the decision and feature dimensions. Our approach and analysis rely on placing a margin around problematic predictions that do not yield unique optimal solutions and then providing generalization bounds in the context of a modified margin SPO loss function that is Lipschitz continuous. Finally, we characterize the strength property and show that the modified SPO loss can be computed efficiently for both strongly convex bodies and polytopes with an explicit extreme point representation. Funding: O. El Balghiti thanks Rayens Capital for their support. A. N. Elmachtoub acknowledges the support of the National Science Foundation (NSF) [Grant CMMI-1763000]. P. Grigas acknowledges the support of NSF [Grants CCF-1755705 and CMMI-1762744]. A. Tewari acknowledges the support of the NSF [CAREER grant IIS-1452099] and a Sloan Research Fellowship.

Disjoint Bilinear Optimization: A Two-Stage Robust Optimization Perspective

In this paper, we focus on a subclass of quadratic optimization problems, that is, disjoint bilinear optimization problems. We first show that disjoint bilinear optimization problems can be cast as two-stage robust linear optimization problems with fixed-recourse and right-hand-side uncertainty, which enables us to apply robust optimization techniques to solve the resulting problems. To this end, a solution scheme based on a blending of three popular robust optimization techniques is proposed. For disjoint bilinear optimization problems with a polyhedral feasible region and a general convex feasible region, we show that, under mild regularity conditions, the convex relaxations of the original bilinear formulation and its two-stage robust reformulation obtained from a reformulation-linearization-based technique and linear decision rules, respectively, are equivalent. For generic bilinear optimization problems, the convex relaxations from the reformulation-linearization-based technique are generally tighter than the one from linear decision rules. Numerical experiments on bimatrix games, synthetic disjoint bilinear problem instances, and convex maximization problems demonstrate the efficiency and effectiveness of the proposed solution scheme. Summary of Contribution: Computing solutions for disjoint bilinear optimization problems are of much interest in real-life applications, yet they are, in general, computationally intractable. This paper proposes a computationally tractable approximation as well as a convergent algorithm to the optimal values of such problems. Extensive computational experiments on (i) (constrained) bimatrix games, (ii) synthetic disjoint bilinear problems, and (iii) convex maximization problems are conducted to demonstrate the effectiveness and efficiency of the proposed approach.

Optimal Trajectory Generation for Intelligent Vehicles in Complex Traffic Based on Iteration Convex Optimization

Intelligent vehicles face considerable challenges in the complex traffic environment since they need to deal with various constraints and elements. This dissertation puts forward a novel trajectory planning framework for intelligent vehicles to generate safe and optimal driving trajectories. First, we design a spatiotemporal occupancy framework to deal with all kinds of elements in the complex driving environment based on the Frenét frame. This framework unifies various constraints on the road in the three-dimensional spatiotemporal representation and clearly describes the collision-free configuration space. Then we use the convex approximation method to construct a time-varying convex feasible region based on the above accurate temporal and spatial description. We formulate the trajectory planning problem as a standard quadratic programming formulation with collision-free and dynamics constraints. Finally, we apply the iterative convex optimization algorithm to solve the quadratic programming problem in the time-varying convex feasible region. Moreover, we design several typical experimental scenarios and have verified that the proposed method has good effectiveness and real-time.

Nonlinear entanglement witnesses for four qubits in mutually unbiased bases

Entanglement witness is a Hermitian operator that is useful for detecting the genuine multipartite entanglement of mixed states. Nonlinear entanglement witnesses have the advantage of a wider detection range in the entangled region. We construct genuine entanglement witnesses for four qubits density matrices in the mutually unbiased basis. First, we find the convex feasible region with positive partial transpose states. Then, to reveal the entangled regions, we present some appropriate linear entanglement witnesses and, we find the envelope of this family of linear witnesses as a nonlinear witness. Finally, we study thermal entanglement and we show for some Hamiltonians the witnesses can detect the entanglement at all temperatures.

A Two-Stage Mathematical Programming Approach for the Solution of Combined Heat and Power Economic Dispatch

The optimal generation scheduling of combined heat and power units aims to minimize total cost of generation units satisfying power and heat demands.The combined heat and power economic dispatch problem should be solved considering a series of electrical and operational constraints, which will be more complicated considering valve-point effects of power-only units and nonconvex feasible operating regions of cogeneration units. In such a case, the problem has nonconvex and nondifferential characteristic. In this article, a two-stage model is proposed to handle the nonconvexity and nondifferentiability of valve-point effects in cost functions of power-only units. The proposed model obtains a convex feasible operating region in the first stage using a linear approximation model. In addition, an equivalent formulation is proposed in the second stage to handle the nondifferentiable term of valve-point effects using the obtained convex feasible operating regions. The proposed model is implemented on two case studies to evaluate performance of the proposed model.

An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.

Dynamic pricing with capacity constraints and inventory replenishment

This paper describes a fast algorithm for solving a capacitated dynamic pricing problem where the producer has the ability to store inventory. The pricing problem described is a quadratic programming problem with a structure that can be solved eectively by a dual algorithm. The proposed algorithm gives a solution satisfying the Karush-Kuhn-Tucker conditions. This, combined with the fact that the problem has a convex feasible region with a concave objective function which we want to maximize, implies that the proposed algorithm gives a globally optimal solution. The algorithm is illustrated by numerical examples for both the single-item and the multi-item cases.

Maximizing for the sum of ratios of two convex functions over a convex set

This paper presents an algorithm for globally maximizing a sum of convex–convex ratios problem with a convex feasible region, which does not require involving all the functions to be differentiable and requires that their sub-gradients can be calculated efficiently. To our knowledge, little progress has been made for globally solving this problem so far. The algorithm uses a branch and bound scheme in which the main computational effort involves solving a sequence of linear programming subproblems. Because of these properties, the algorithm offers a potentially attractive means for globally solving the sum of convex–convex ratios problem over a convex feasible region. It has been proved that the algorithm possesses global convergence. Finally, the numerical experiments are given to show the feasibility of the proposed algorithm.

Using conical partition to globally maximizing the nonlinear sum of ratios

This article presents a global optimization algorithm for globally maximizing the sum of concave–convex ratios problem with a convex feasible region. The algorithm uses a branch and bound scheme where a concave envelope of the objective function is constructed to obtain an upper bound of the optimal value by using conical partition. As a result, the upper-bound subproblems during the algorithm search are all ordinary convex programs with less variables and constraints and do not grow in size from iterations to iterations in the computation procedure, and furthermore a new bounding tightening strategy is proposed such that the upper-bound convex relaxation subproblems are closer to the original nonconvex problem to enhance solution procedure. At last, some numerical examples are given to vindicate our conclusions.

Motion Planning in Obstacle Rich Environments

In this paper we present a multi-layer approach for motion planning in obstacle rich environments.The approach is built on the principle of separation of concern which partitions the motion planning problem into multiple independent layers. This enables design space exploration at each layer. We partition the motion planning algorithm into a roadmap layer and an optimal control layer. Elements of computational geometry are used to process the obstacle rich environment and generate a set of convex feasible regions, which is then used by the optimal control layer to generate trajectories while satisfying dynamics of the vehicle. The roadmap layer ignores the dynamics of the system, and plans paths at a global level using coarse representation of the environment. The optimal control layer ignores the complexity of the environment and plans paths at a mid-level using fine representation of the dynamics and the environment. In this manner a separation of concern is achieved. This decomposition enables computationally tractable methods to be developed for addressing motion planning in complex environments.

An interactive weight space reduction procedure for nonlinear multiple objective mathematical programming

To make a decision that is defined by multiple, conflicting objectives it is necessary to know the relative importance of the different objectives. In this paper we present an interactive method and the underlying theory for solving multiple objective mathematical programming problems defined by a convex feasible region and concave, continuously differentiable objective functions. The relative importance of the different objectives for a decision maker is elicited by using binary comparisons of objective function vectors. The method is cognitively easy to use and in test problems has rapidly converged to an optimal solution.

Global optimization problems and domain reduction strategies

In this paper we discuss domain reduction strategies for global optimization problems with a nonconvex objective function over a bounded convex feasible region. After introducing a standard domain reduction and its iterated version, we will introduce a new reduction strategy. Under mild assumptions, we will prove the equivalence between the new domain reduction and the iterated version of the standard one, allowing a new interpretation of the latter and a new way of computing it. Finally, we prove that any “reasonable” domain reduction strategy is independent of the order by which variables are processed.

On feasible regions of lamination parameters for lay-up optimization of laminated composites

The stiffness tensors of a laminated composite may be expressed as a linear function of material invariants and lamination parameters. Owing to the nature of orienting unidirectional laminae ply by ply, lamination parameters, which are trigonometric functions of the ply orientation, are interrelated. In optimization studies, lamination parameters are often treated as independent design variables constrained by inequality relationships to feasible regions that depend on their values. The relationships between parameters enclose a convex feasible region of lamination parameters which is generally unknown. The convexity properties allow the efficient optimization of laminated composite structures where lamination parameters are used as design variables. Herein, a two-level method is presented to determine the feasible regions of lamination parameters where potential ply orientations are a predefined finite set. At the first level, the feasible region of the in-plane, coupling and out-of-plane lamination parameters is determined separately using convex hulls. At the second level, a nonlinear algebraic identity is used to relate the in-plane, coupling and out-of-plane lamination parameters to each other. This general approach yields all constraints on the feasible regions of lamination parameters for a predefined set of ply orientations.

Boundary search and simplex decomposition method for MDO problems with a convex or star-like state parameter region

One major challenge in multidisciplinary design optimization (MDO) is the presence of couplings among state parameters, which demands an iterative and often expensive system analysis (SA) process for each function evaluation in optimization. This paper offers a new perspective and proposes a corresponding method for solving MDO problems. The proposed method, named the boundary search and simplex decomposition method (BSSDM), geometrically captures the relation among coupled state parameters with a feasible state parameter region. Given the feasible state parameter region, the SA can be avoided during the optimization of the system objective function. To identify the feasible state parameter region, a search strategy is developed to find boundary points of the region. In the boundary search process, a collaboration model (CM) is applied to maintain the feasibility of samples with respect to the SA. In search of the system optimum in the feasible region, a robust simplex decomposition algorithm is developed for convex and star-like feasible state parameter regions. The BSSDM is tested with two numerical cases, one of which is an MDO problem constrained by a convex state parameter region, and the other is a SA problem with a star-like state parameter region. All results are then validated, and the results show the promising capability of the proposed BSSDM.

A geometry constraint handling technique for stiffener layout optimization problem

Beam stiffeners have frequently been used for raising natural frequencies of base structures. In stiffener layout optimization problems, most of the previous researches considering the position and/or the length of the stiffener as design variables dealt with structures having just simple convex shapes such as a square or rectangle. The reason was because concave shape structures have difficulties in formulating geometry constraints. In this paper, a new geometry constraint handling technique, which can define both convex and concave feasible regions and measure a degree of geometry constraint violation, was proposed. Evolution strategies (ESs) was utilized as an optimization tool. In addition, the constraint-handling technique of EVOSLINOC (EVOlution Strategies for scalar optimization with LInear and NOnlinear Constraints) was utilized to solve constrained optimization problems. From numerical examples, the proposed geometry constraint handling technique was verified and proved that the technique can easily be applied to structures in not only convex but also concave shapes, even with a protrusion or interior holes.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush-Kuhn-Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

Connectedness of the Efficient Set for Strictly Quasiconcave Sets

Given a closed subset X in , we show the connectedness of its efficient points or nondominated points when X is sequentially strictly quasiconcave. In the particular case of a maximization problem with n continuous and strictly quasiconcave objective functions on a compact convex feasible region of , we deduce the connectedness of the efficient frontier of the problem. This work solves the open problem of the efficient frontier for strictly quasiconcave vector maximization problems.

Robust bounds and throughput guarantees for closed multiclass queueing networks

To use queueing theory for analyzing real computing systems, we may make assumptions that are strictly speaking untrue. The problem is especially severe for multiclass systems with widely differing service times. This paper provides an exact analysis for bounds for systems with greatly relaxed assumptions. Service times can have arbitrary NBUE distributions, different by class even at FIFO nodes. Routing can be arbitrary, including dependencies along the route, provided the number of visits to a device per response cycle is random with a known expectation. Only the mean service time and mean visit rates at nodes need to be specified. A new lower throughput bound is found which gives a minimum guaranteed throughput for each class; together with the familiar multiclass asymptotic upper bounds they give a convex feasible region in a multidimensional throughput space. A detailed analysis is given for queueing network models of systems with infinite-server nodes as well as queueing nodes with various service disciplines: FIFO, processor sharing, and priority (preemptive as well as non-preemptive) scheduling. Because the feasible region may be a complicated shape which is difficult to visualize, the results can be re-interpreted as a set of bounds on the separate throughputs. This is equivalent to a circumscribed rectangular region called the “robust box bounds”. Computation of these bounds is carried out by a novel technique based on interval arithmetic as implemented in BNR Prolog language. A method for computing approximate system throughput from the box bounds is also proposed in the paper. Using Little's law and utilization law with the bounds on throughput bounds on response times and device utilizations are obtained. These analytic techniques can be effectively utilized for analyzing the performance of distributed systems as well as other types of computing systems.

Equivalent primal and dual differentiable reformulations of the Euclidean multifacility location problem

In this paper, we consider two equivalent differentiable reformulations of the nondifferentiable Euclidean multifacility location problem (EMFLP). The first of these is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). The resulting formulation turns out to be identical to the known dual problem proposed by Francis and Cabot [1]. Hence, besides providing an easy direct derivation of the dual problem, this approach lends insights into its connections with classical Lagrangian duality and related results. In particular, it characterizes a straightforward recovery of primal location decisions. The second equivalent differentiable formulation is constructed directly in the primal space. Although the individual constraints of the resulting problem are generally nonconvex, we show that their intersection represents a convex feasible region. We then establish the relationship between the Karush–Kuhn–Tucker (KKT) conditions for this problem and the necessary and sufficient optimality conditions for EMFLP. This lends insights into the possible performance of standard differentiable nonlinear programming algorithms when applied to solve this reformulated problem. Some computational results on test problems from the literature, and other randomly generated problems, are also provided.

Convergence Properties of Minimization Algorithms for Convex Constraints Using a Structured Trust Region

In this paper, we present a class of trust region algorithms for minimization problems within convex feasible regions in which the structure of the problem is explicitly used in the definition of the trust region. This development is intended to reflect the possibility that some parts of the problem may be more accurately modelled than others, a common occurrence in large-scale nonlinear applications. After describing the structured trust region mechanism, we prove global convergence for all algorithms in our class.