Knapsack problems with sigmoid utilities: Approximation algorithms via hybrid optimization

We study a class of non-convex optimization problems involving sigmoid functions. We show that sigmoid functions impart a combinatorial element to the optimization variables and make the global optimization computationally hard. We formulate versions of the knapsack problem, the generalized assignment problem and the bin-packing problem with sigmoid utilities. We merge approximation algorithms from discrete optimization with algorithms from continuous optimization to develop approximation algorithms for these NP-hard problems with sigmoid utilities.     

Elliptic cone optimization and primal–dual path-following algorithms

In elliptic cone optimization problems, we minimize a linear objective function over the intersection of an affine linear manifold with the Cartesian product of the so-called elliptic cones. We present some general classes of optimization problems that can be cast as elliptic cone programmes such as second-order cone programmes and circular cone programmes. We also describe some real-world applications of this class of optimization problems. We study and analyse the Jordan algebraic structure of the elliptic cones. Then, we present a glimpse of the duality theory associated with elliptic cone optimization. A primal–dual path-following interior-point algorithm is derived for elliptic cone optimization problems. We prove the polynomial convergence of the proposed algorithms by showing that the logarithmic barrier is a strongly self-concordant barrier. The numerical examples show the path-following algorithms are efficient.     

Scalarization method for Levitin–Polyak well-posedness of vectorial optimization problems

In this paper, we develop a method of study of Levitin?CPolyak well-posedness notions for vector valued optimization problems using a class of scalar optimization problems. We first introduce a non-linear scalarization function and consider its corresponding properties. We also introduce the Furi?CVignoli type measure and Dontchev?CZolezzi type measure to scalar optimization problems and vectorial optimization problems, respectively. Finally, we construct the equivalence relations between the Levitin?CPolyak well-posedness of scalar optimization problems and the vectorial optimization problems.     

A global optimization method for the design of space trajectories

The problem of optimally designing a trajectory for a space mission is considered in this paper. Actual mission design is a complex, multi-disciplinary and multi-objective activity with relevant economic implications. In this paper we will consider some simplified models proposed by the European Space Agency as test problems for global optimization (GTOP database). We show that many trajectory optimization problems can be quite efficiently solved by means of relatively simple global optimization techniques relying on standard methods for local optimization. We show in this paper that our approach has been able to find trajectories which in many cases outperform those already known. We also conjecture that this problem displays a “funnel structure” similar, in some sense, to that of molecular optimization problems.     

Global optimization by continuous grasp

We introduce a novel global optimization method called Continuous GRASP (C-GRASP) which extends Feo and Resende’s greedy randomized adaptive search procedure (GRASP) from the domain of discrete optimization to that of continuous global optimization. This stochastic local search method is simple to implement, is widely applicable, and does not make use of derivative information, thus making it a well-suited approach for solving global optimization problems. We illustrate the effectiveness of the procedure on a set of standard test problems as well as two hard global optimization problems.     

Integral global minimization: Algorithms,implementations and numerical tests

The theoretical foundation of integral global optimization has become widely known and well accepted [4],[24],[25]. However, more effort is needed to demonstrate the effectiveness of the integral global optimization algorithms. In this work we detail the implementation of the integral global minimization algorithms. We describe how the integral global optimization method handles nonconvex unconstrained or box constrained, constrained or discrete minimization problems. We illustrate the flexibility and the efficiency of integral global optimization method by presenting the performance of algorithms on a collection of well known test problems in global optimization literature. We provide the software which solves these test problems and other minimization problems. The performance of the computations demonstrates that the integral global algorithms are not only extremely flexible and reliable but also very efficient.Research supported partially by NSERC grant and Mount St Vincent University research grant.     

Portfolio optimization with linear and fixed transaction costs

We consider the problem of portfolio selection, with transaction costs and constraints on exposure to risk. Linear transaction costs, bounds on the variance of the return, and bounds on different shortfall probabilities are efficiently handled by convex optimization methods. For such problems, the globally optimal portfolio can be computed very rapidly. Portfolio optimization problems with transaction costs that include a fixed fee, or discount breakpoints, cannot be directly solved by convex optimization. We describe a relaxation method which yields an easily computable upper bound via convex optimization. We also describe a heuristic method for finding a suboptimal portfolio, which is based on solving a small number of convex optimization problems (and hence can be done efficiently). Thus, we produce a suboptimal solution, and also an upper bound on the optimal solution. Numerical experiments suggest that for practical problems the gap between the two is small, even for large problems involving hundreds of assets. The same approach can be used for related problems, such as that of tracking an index with a portfolio consisting of a small number of assets.     

Scalarization and pointwise well-posedness in vector optimization problems

The aim of this paper is applying the scalarization technique to study some properties of the vector optimization problems under variable domination structure. We first introduce a nonlinear scalarization function of the vector-valued map and then study the relationships between the vector optimization problems under variable domination structure and its scalarized optimization problems. Moreover, we give the notions of DH-well-posedness and B-well-posedness under variable domination structure and prove that there exists a class of scalar problems whose well-posedness properties are equivalent to that of the original vector optimization problem.     

DC approach to weakly convex optimization and nonconvex quadratic optimization problems

Motivated by weakly convex optimization and quadratic optimization problems, we first show that there is no duality gap between a difference of convex (DC) program over DC constraints and its associated dual problem. We then provide certificates of global optimality for a class of nonconvex optimization problems. As an application, we derive characterizations of robust solutions for uncertain general nonconvex quadratic optimization problems over nonconvex quadratic constraints.     

Multiobjective GRASP with Path Relinking

In this paper we review and propose different adaptations of the GRASP metaheuristic to solve multiobjective combinatorial optimization problems. In particular, we describe several alternatives to specialize the construction and improvement components of GRASP when two or more objectives are considered. GRASP has been successfully coupled with Path Relinking for single-objective optimization. Moreover, we propose different hybridizations of GRASP and Path Relinking for multiobjective optimization. We apply the proposed GRASP with Path Relinking variants to two combinatorial optimization problems, the biobjective orienteering problem and the biobjective path dissimilarity problem. We report on empirical tests with 70 instances and 30 algorithms, that show that the proposed heuristics are competitive with the state-of-the-art methods for these problems.     

What makes an optimization problem hand?

We address the question: “Are some classes of combinatorial optimization problems intrinsically harder than others, without regard to the algorithm one uses, or can difficulty only be assessed relative to particular algorithms?” We provide a measure of the hardness of a particular optimization problem for a particular optimization algorithm and present two algorithm-independent quantities that use this measure to provide answers to our question. In the first of these we average hardness over all possible algorithms and show that according to this quantity, there are no distinctions between optimization problems. In this sense no problems are intrinsically harder than others. For the second quantity, rather than average over all algorithms, we consider the level of hardness of a problem (or class of problems) for the associated optimal algorithm. By this criteria there are classes of problems that are intrinsically harder than others.     

Duality Theory in Fuzzy Optimization Problems

A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions.     

一个标量泛函的研究及其在集值优化问题中的应用

本文在K条件下,研究了所给标量泛函的连续性和拟凸性,并利用该标量泛函,将集值优化问题转化为均衡问题,进而研究了含约束的集值优化问题弱充分解的存在性和拟集值优化问题强逼近解映射的上半连续性与下半连续性.与最近的文献相比,我们的方法是新的,条件和结论也更具一般性.     

An unbounded Sum-of-Squares hierarchy integrality gap for a polynomially solvable problem

In this expository article, we study optimization problems specified via linear and relative entropy inequalities. Such relative entropy programs (REPs) are convex optimization problems as the relative entropy function is jointly convex with respect to both its arguments. Prominent families of convex programs such as geometric programs (GPs), second-order cone programs, and entropy maximization problems are special cases of REPs, although REPs are more general than these classes of problems. We provide solutions based on REPs to a range of problems such as permanent maximization, robust optimization formulations of GPs, and hitting-time estimation in dynamical systems. We survey previous approaches to some of these problems and the limitations of those methods, and we highlight the more powerful generalizations afforded by REPs. We conclude with a discussion of quantum analogs of the relative entropy function, including a review of the similarities and distinctions with respect to the classical case. We also describe a stylized application of quantum relative entropy optimization that exploits the joint convexity of the quantum relative entropy function.     

Calmness and Exact Penalization in Constrained Scalar Set-Valued Optimization

In this paper, we study a class of constrained scalar set-valued optimization problems, which includes scalar optimization problems with cone constraints as special cases. We introduce (local) calmness of order??? for this class of constrained scalar set-valued optimization problems. We show that the (local) calmness of order??? is equivalent to the existence of a (local) exact set-valued penalty map.     

目标函数扰动的集值优化问题的稳定性

本文研究了由目标函数扰动的集值优化问题的有效点集所定义的集值映射的半连续性.讨论了目标函数扰动的集值优化问题在上半连续意义下的稳定性.特别地,在广义适定性条件下,证明了集值优化问题在上半连续意义下的稳定性.     

Existence of a Saddle Point in Nonconvex Constrained Optimization

The existence of a saddle point in nonconvex constrained optimization problems is considered in this paper. We show that, under some mild conditions, the existence of a saddle point can be ensured in an equivalent p-th power formulation for a general class of nonconvex constrained optimization problems. This result expands considerably the class of optimization problems where a saddle point exists and thus enlarges the family of nonconvex problems that can be solved by dual-search methods.     

Robust assortment optimization using worst-case CVaR under the multinomial logit model

We consider robust assortment optimization problems with partial distributional information of parameters in the multinomial logit choice model. The objective is to find an assortment that maximizes a revenue target using a distributionally robust chance constraint, which can be approximated by the worst-case Conditional Value-at-Risk. We show that our problems are equivalent to robust assortment optimization problems over special uncertainty sets of parameters, implying the optimality of revenue-ordered assortments under certain conditions.     

Optimization with multivariate stochastic dominance constraints

We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.