General Class Of Optimization Problems Research Articles

Metric Optimization in Penner Coordinates

Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of optimization problems involving metrics, including optimization and interpolation, while retaining the key solution existence guarantees available for discrete conformal maps.

Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds

In this paper, we consider a class of multiobjective mathematical programming problems with equilibrium constraints on Hadamard manifolds (in short, (MMPEC)). We introduce the generalized Guignard constraint qualification for (MMPEC) and employ it to derive Karush–Kuhn–Tucker (KKT)-type necessary optimality criteria. Further, we derive sufficient optimality criteria for (MMPEC) using geodesic convexity assumptions. The significance of the results deduced in the paper has been demonstrated by suitable non-trivial examples. The results deduced in this article generalize several well-known results in the literature to a more general space, that is, Hadamard manifolds, and extend them to a more general class of optimization problems. To the best of our knowledge, this is the first time that generalized Guignard constraint qualification and optimality conditions have been studied for (MMPEC) in manifold settings.

A Convex Reformulation and an Outer Approximation for a Large Class of Binary Quadratic Programs

Binary Quadratic Program with Variable Partitioning Constraints The binary quadratic program with variable partitioning constraints is a very general class of optimization problems that is very difficult to solve because of the nonconvexity and integrality of the variables and is ubiquitous, among others, in network design, computer vision, and transportation and logistics. In their article, “A convex reformulation and an outer approximation for a large class of binary quadratic programs,” Rostami et al. show how to transform such a nonconvex challenging problem into a convex bilinear program with decomposable structure. The authors develop a branch-and-cut algorithm based on outer approximation cuts, in which the cuts are generated on the fly by efficiently solving separation subproblems. The results of their computational experiments on different problems confirm the efficacy of the solution methods in solving large-scale problem instances.

A Novel Active Learning Regression Framework for Balancing the Exploration-Exploitation Trade-Off.

Recently, active learning is considered a promising approach for data acquisition due to the significant cost of the data labeling process in many real world applications, such as natural language processing and image processing. Most active learning methods are merely designed to enhance the learning model accuracy. However, the model accuracy may not be the primary goal and there could be other domain-specific objectives to be optimized. In this work, we develop a novel active learning framework that aims to solve a general class of optimization problems. The proposed framework mainly targets the optimization problems exposed to the exploration-exploitation trade-off. The active learning framework is comprehensive, it includes exploration-based, exploitation-based and balancing strategies that seek to achieve the balance between exploration and exploitation. The paper mainly considers regression tasks, as they are under-researched in the active learning field compared to classification tasks. Furthermore, in this work, we investigate the different active querying approaches—pool-based and the query synthesis—and compare them. We apply the proposed framework to the problem of learning the price-demand function, an application that is important in optimal product pricing and dynamic (or time-varying) pricing. In our experiments, we provide a comparative study including the proposed framework strategies and some other baselines. The accomplished results demonstrate a significant performance for the proposed methods.

Approximate Multiparametric Mixed-Integer Convex Programming

We propose an algorithm for generating explicit solutions of multiparametric mixed-integer convex programs to within a given suboptimality tolerance. The algorithm is applicable to a very general class of optimization problems, but is most useful for hybrid model predictive control, where online implementation is hampered by the worst-case exponential complexity of mixed-integer solvers. The output is a simplicial partition which defines a static map from the current state to a suboptimal solution. The primary theoretical contribution of this letter is to introduce a non-zero optimal cost overlap metric which is necessary and sufficient for convergence. The overlap size is also linked to partition complexity. The algorithm is massively parallelizable and our implementation, which is publicly available, is run on a cluster of several hundred processors. Not only does our solution have a deterministic runtime, simulations show that our approach is faster than online optimization by up to three orders of magnitude.

Recursive Contracts

We obtain a recursive formulation for a general class of optimization problems with forward‐looking constraints which often arise in economic dynamic models, for example, in contracting problems with incentive constraints or in models of optimal policy. In this case, the solution does not satisfy the Bellman equation. Our approach consists of studying a recursive Lagrangian. Under standard general conditions, there is a recursive saddle‐point functional equation (analogous to a Bellman equation) that characterizes a recursive solution to the planner's problem. The recursive formulation is obtained after adding a co‐state variable μ t summarizing previous commitments reflected in past Lagrange multipliers. The continuation problem is obtained with μ t playing the role of weights in the objective function. Our approach is applicable to characterizing and computing solutions to a large class of dynamic contracting problems.

Model-based capacitated clustering with posterior regularization

We propose a heuristic approach to address the general class of optimization problems involving the capacitated clustering of observations consisting of variable values that are realizations from respective probability distributions. Based on the expectation-maximization algorithm, our approach unifies Gaussian mixture modeling for clustering analysis and cluster capacity constraints using a posterior regularization framework. To test our algorithm, we consider the capacitated p-median problem in which the observations consist of geographic locations of customers and the corresponding demand of these customers. Our heuristic has superior performance compared to classic geometrical clustering heuristics, with robust performance over a collection of instance types.

Optimal portfolio trading subject to stochastic dominance constraints under second‐order autoregressive price dynamics

AbstractThis paper studies the optimal portfolio trading problem under the generalized second‐order autoregressive execution price model. The problem of minimizing expected execution cost under the proposed price model is formulated as a quadratic programming (QP) problem. For a risk‐averse trader, problem formulation under the second‐order stochastic dominance constraints results in a quadratically constrained QP problem. Under some conditions on the execution price model, it is proved that the portfolio trading problems for risk‐neutral and risk‐averse traders become convex programming problems, which have many theoretical and computational advantages over the general class of optimization problems. Extensive numerical illustrations are provided, which render the practical significance of the proposed execution price model and the portfolio trading problems.

Solving the Minimum-Time Velocity Planning Problem through an Hypergraph-Based Approach

In a previous work an algorithm with linear-time computational complexity with respects to the number of variables is presented, providing an optimal solution for the minimum-time velocity planning problem. One limitation of such work is that the obtained velocity profile is not sufficiently smooth. In this work we try to obtain a smoother velocity profile, adding additional constraints on the absolute value of the second derivative of velocity with respect to the arc-length. We propose an algorithm that is able to efficiently solve the minimum-time velocity planning problem when only the lower bound of the second derivative of the velocity is considered. We will also see that the minimum time velocity planning problem under consideration belongs to a more general class of optimization problems, which can be tackled by the same approach. The approach is illustrated through an example and tested over a set of randomly generated instances. Properties of the proposed algorithm are proved.

A model problem for Mean Field Games on networks

In [14], Gueant, Lasry and Lions considered the model problem ``What time does meeting start?'' as a prototype for a general class of optimization problems with a continuum of players, called Mean Field Games problems. In this paper we consider a similar model, but with the dynamics of the agents defined on a network. We discuss appropriate transition conditions at the vertices which give a well posed problem and we present some numerical results.

Spanning cactus of a graph: Existence, extension, optimization, and approximation

We show that testing if an undirected graph contains a bridgeless spanning cactus is NP-hard. As a consequence, the minimum spanning cactus problem (MSCP) on an undirected graph with 0–1 edge weights is NP-hard. For any subgraph S of Kn, we give polynomially testable necessary and sufficient conditions for S to be extendable to a cactus in Kn and the weighted version of this problem is shown to be NP-hard. A spanning tree is shown to be extendable to a cactus in Kn if and only if it has at least one node of even degree. When S is a spanning tree, we show that the weighted version can also be solved in polynomial time. Further, we give an O(n3) algorithm for computing a minimum cost spanning tree with at least one vertex of even degree on a graph on n nodes. Finally, we show that for a complete graph with edge-costs satisfying the triangle inequality, the MSCP is equivalent to a general class of optimization problems that properly includes the traveling salesman problem and they all have the same approximation hardness.

A reformulation framework for global optimization

In this paper, we present a global optimization method for solving nonconvex mixed integer nonlinear programming (MINLP) problems. A convex overestimation of the feasible region is obtained by replacing the nonconvex constraint functions with convex underestimators. For signomial functions single-variable power and exponential transformations are used to obtain the convex underestimators. For more general nonconvex functions two versions of the so-called ?BB-underestimator, valid for twice-differentiable functions, are integrated in the actual reformulation framework. However, in contrast to what is done in branch-and-bound type algorithms, no direct branching is performed in the actual algorithm. Instead a piecewise convex reformulation is used to convexify the entire problem in an extended variable-space, and the reformulated problem is then solved by a convex MINLP solver. As the piecewise linear approximations are made finer, the solution to the convexified and overestimated problem will form a converging sequence towards a global optimal solution. The result is an easily-implementable algorithm for solving a very general class of optimization problems.

On Linear Programming Duality and Necessary and Sufficient Conditions in Minimax Theory

In this paper we discuss necessary and sufficient conditions for different minimax results to hold using only linear programming duality and the finite intersection property for compact sets. It turns out that these necessary and sufficient conditions have a clear interpretation within zero-sum game theory. We apply these results to derive necessary and sufficient conditions for strong duality for a general class of optimization problems.

Arborescence optimization problems solvable by Edmonds’ algorithm

We consider a general class of optimization problems regarding spanning trees in directed graphs (arborescences). We present an algorithm for solving such problems, which can be considered as a generalization of Edmonds’ algorithm for the solution of the minimum sum arborescence problem. The considered class of optimization problems includes as special cases the standard minimum sum arborescence problem, the bottleneck and lexicographically optimal arborescence problem, as well as the widest-minimum sum arborescence problem.

On bounds for the existence of a bistable controller

By S(r) we shall denote the set of polynomials nonzero on the closed disk {z : |z|⩽r} . The problem to be studied is given as follows. Avoidance Problem. Given polynomials A and B, nonnegative integers n, m, and positive numbers r 1, r 2, r 3, determine if there exist P∈ S(r 1) , Q∈ S(r 2) with degrees n and m, respectively, such that AP+ BQ is in S(r 3) . It is shown that the Avoidance Problem for r 1= r 2= r 3=1 is equivalent to the well-known Bistable Stabilization Problem for a large class of control systems. Also, a general class of optimization problems for solving the Avoidance Problem is introduced. The class of optimization problems includes problems with sets more general than disks S(r) . In addition, an algorithm is proposed for solving these optimization problems, and numerical experience is reported. As an application, bistable controllers for the plant P β(z)=z/(z 2+β) are shown to exist for values of β as low as 0.0246, thus improving on previously reported results by Blondel et al. for this difficult case. Furthermore, it is shown how in some cases it is possible to find optimal controllers by using a combination of numerical and algebraic techniques, and a variant of the algorithm already introduced. As an example, a low order controller is found for P β for the case when β=7−4 3 ≈0.0717 . This method could be applied to other problems where the plant depends on a parameter.

Optimal investment with transaction costs and without semimartingales

We consider a general class of optimization problems in financial markets with incomplete information and transaction costs. Under a no-arbitrage condition strictly weaker than the existence of a martingale measure, and when asset prices are quasi-left-continuous processes, we show the existence of optimal strategies. Applications include maximization of expected utility, minimization of coherent risk measures and hedging of contingent claims.

Disjunctive optimization, max-separable problems and extremal algebras

The paper was motivated by solution methods suggested in the literature for solving linear optimization problems over (max,+)- or (max,min)-algebras and certain class of so called max-separable optimization problems. General features of these optimization problems, which play a crucial role in the optimization methods were used to formulate a general class of optimization problems with disjunctive constraints and a max-separable objective function and suggest a solution procedure for solving such problems. Linear problems over (max,+)-algebras and the max-separable problems are contained in this general class of optimization problems as special cases.

Simultaneous optimization and solution of systems described by differential/algebraic equations

Engineering approaches to the solution of constrained variational problems often involve converting the problem into a nonlinear programming (NLP) problem and solving it using current NLP methods. These methods usually use a sequential optimization and solution strategy. We propose a method, using piecewise constant functions for the independent variables, that combines the technologies of quasi-Newton optimization algorithms and global spline collocation to simultaneously optimize and integrate systems described by differential/algebraic equations. A computer implementable algorithm is discussed and three test problems are solved. The algorithm allows the solution of a more general class of optimization problems than previous methods employing this strategy.

Constrained Optimization of Functionals with Search Theory Applications

We find necessary and sufficient conditions for maximizing a wide class of nonlinear, nonseparable functionals under separable constraints. The crucial restriction on the functionals is that they have a Gateaux differential which is a linear functional with a kernel. The conditions obtained can be applied to a large variety of optimal search problems involving moving targets when effort is infinitely divisible in space. Moreover, the conditions have been used to construct very efficient algorithms for solving these problems. It is conjectured that these results are useful in a general class of optimization problems that extend well beyond the search theory examples presented in this paper.

Decentralized optimization for distributed-lag models of discrete systems

The approach discussed in this paper solves a general class of optimization problems for discrete dynamic systems which include distributed lag, distributed and/or multiple pure delays, and constraints both in state and control variables. The overall system equation of this problem is described by a multidimensional nonlinear difference equation of high-order which is called the distributed-lag model. Applying Lagrange duality theory to the original problem, the dual problem is formulated, and the decomposition of the optimization process in stage is obtained. It is shown that by solving the dual problem the delay terms can be easily handled and the optimal solution to the original problem is obtained without reducing the multi-dimensional high-order system equation to a conventional larger dimensional first-order system equation. It is also shown that the dual decentralized method in this paper is easier to cope with state and control constraints than the primal method in the space of control, i.e. gradient and other techniques. The approach developed in this paper is compared with other methods using a simple example, and is applied to a combined marketing and production control problem. Some computational results are included.