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Cooperative games form an important class of problems in game theory, where a key goal is to distribute a value among a set of players who are allowed to cooperate by forming coalitions. An outcome of the game is given by an allocation vector that assigns a value share to each player. A crucial aspect of such games is submodularity (or convexity). Indeed, convex instances of cooperative games exhibit several nice properties, e.g. regarding the existence and computation of allocations realizing some of the most important solution concepts proposed in the literature. For this reason, a relevant question is whether one can give a polynomial-time characterization of submodular instances, for prominent cooperative games that are in general non-convex. In this paper, we focus on a fundamental and widely studied cooperative game, namely the spanning tree game. An efficient recognition of submodular instances of this game was not known so far, and explicitly mentioned as an open question in the literature. We here settle this open problem by giving a polynomial-time characterization of submodular spanning tree games.

We propose a complete-search algorithm for solving a class of non-convex, possibly infinite-dimensional, optimization problems to global optimality. We assume that the optimization variables are in a bounded subset of a Hilbert space, and we determine worst-case run-time bounds for the algorithm under certain regularity conditions of the cost functional and the constraint set. Because these run-time bounds are independent of the number of optimization variables and, in particular, are valid for optimization problems with infinitely many optimization variables, we prove that the algorithm converges to an \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document}ε-suboptimal global solution within finite run-time for any given termination tolerance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0. Finally, we illustrate these results for a problem of calculus of variations.

This paper discusses an outer-approximation guided optimization method for constrained neural network inverse problems with rectified linear units. The constrained neural network inverse problems refer to an optimization problem to find the best set of input values of a given trained neural network in order to produce a predefined desired output in presence of constraints on input values. This paper analyzes the characteristics of optimal solutions of neural network inverse problems with rectified activation units and proposes an outer-approximation algorithm by exploiting their characteristics. The proposed outer-approximation guided optimization comprises primal and dual phases. The primal phase incorporates neighbor curvatures with neighbor outer-approximations to expedite the process. The dual phase identifies and utilizes the structure of local convex regions to improve the convergence to a local optimal solution. At last, computation experiments demonstrate the superiority of the proposed algorithm compared to a projected gradient method.

Choice of a risk measure for quantifying risk of an investment portfolio depends on the decision maker’s risk preference. In this paper, we consider the case when such a preference can be described by a law invariant coherent risk measure but the choice of a specific risk measure is ambiguous. We propose a robust spectral risk approach to address such ambiguity. Differing from Wang and Xu (SIAM J Optim 30(4):3198–3229, 2020), the new robust model allows one to elicit the decision maker’s risk preference through pairwise comparisons and use the elicited preference information to construct an ambiguity set of risk spectra. The robust spectral risk measure (RSRM) is based on the worst case risk spectrum from the set. To calculate RSRM and solve the associated optimal decision making problem, we use a technique from Acerbi and Simonetti (Portfolio optimization with spectral measures of risk. Working paper, 2002) to develop a new computational approach which is independent of order statistics and reformulate the robust spectral risk optimization problem as a single deterministic convex programming problem when the risk spectra in the ambiguity set are step-like. Moreover, we propose an approximation scheme when the risk spectra are not step-like and derive a bound for the model approximation error and its propagation to the optimal decision making problems. Some preliminary numerical test results are reported about the performance of the robust model and the computational scheme.

International audience; Sampled-data control systems have steadily been gaining interest for their applications in automatic engineering where they are implemented as digital controllers and recently results have been obtained in optimal control theory for nonlinear sampled-data control systems and certain generalizations. In this paper we derive a Pontryagin maximum principle for general nonlinear finite-dimensional optimal sampled-data control problems with running inequality state constraints. In particular, we obtain a nonpositive averaged Hamiltonian gradient condition with the adjoint vector being a function of bounded variations. Our proof is based on the Ekeland variational principle. In general, optimal control problems with running inequality state constraints are difficult to solve using numerical methods due to the discontinuities (the jumps and the singular part) of the adjoint vector. However in our case we find that under certain general hypotheses the adjoint vector only experiences jumps at most at the sampling times and moreover the trajectory only contacts the running inequality state constraints at most at the sampling times. We call this behavior a bouncing trajectory phenomenon and it constitutes the second major focus of this paper. Finally taking advantage of the bouncing trajectory phenomenon we numerically solve three examples with different kinds of constraints and in several dimensions.

Major advances were recently obtained in the exact solution of vehicle routing problems (VRPs). Sophisticated branch-cut-and-price (BCP) algorithms for some of the most classical VRP variants now solve many instances with up to a few hundreds of customers. However, adapting and reimplementing those successful algorithms for other variants can be a very demanding task. This work proposes a BCP solver for a generic model that encompasses a wide class of VRPs. It incorporates the key elements found in the best existing VRP algorithms: ng-path relaxation, rank-1 cuts with limited memory, path enumeration, and rounded capacity cuts; all generalized through the new concepts of “packing set” and “elementarity set”. The concepts are also used to derive a branching rule based on accumulated resource consumption and to generalize the Ryan and Foster branching rule. Extensive experiments on several variants show that the generic solver has an excellent overall performance, in many problems being better than the best specific algorithms. Even some non-VRPs, like bin packing, vector packing and generalized assignment, can be modeled and effectively solved.

A graph G(V, E) is claw-free if no vertex has three pairwise non-adjacent neighbours. The maximum weight stable set (MWSS) problem in a claw-free graph is a natural generalization of the matching problem and has been shown to be polynomially solvable by Minty and Sbihi in 1980. In a remarkable paper, Faenza, Oriolo and Stauffer have shown that, in a two-step procedure, a claw-free graph can be first turned into a quasi-line graph by removing strips containing all the irregular nodes and then decomposed into {claw, net}-free strips and strips with stability number at most three. Through this decomposition, the MWSS problem can be solved in $$\mathcal{O}(|V|(|V| \log |V| + |E|))$$ time. In this paper, we describe a direct decomposition of a claw-free graph into {claw, net}-free strips and strips with stability number at most three which can be performed in $$\mathcal{O}(|V|^2)$$ time. In two companion papers we showed that the MWSS problem can be solved in $$\mathcal{O}(|E| \log |V|)$$ time in claw-free graphs with $$\alpha (G) \le 3$$ and in $$\mathcal{O}(|V| \sqrt{|E|})$$ time in {claw, net}-free graphs with $$\alpha (G) \ge 4$$ . These results prove that the MWSS problem in a claw-free graph can be solved in $$\mathcal{O}(|V|^2 \log |V|)$$ time, the same complexity of the best and long standing algorithm for the MWSS problem in line graphs.

In this paper we study nonconvex and nonsmooth multi-block optimization over Euclidean embedded (smooth) Riemannian submanifolds with coupled linear constraints. Such optimization problems naturally arise from machine learning, statistical learning, compressive sensing, image processing, and tensor PCA, among others. By utilizing the embedding structure, we develop an ADMM-like primal-dual approach based on decoupled solvable subroutines such as linearized proximal mappings, where the duality is with respect to the embedded Euclidean spaces. First, we introduce the optimality conditions for the afore-mentioned optimization models. Then, the notion of $$\epsilon $$ -stationary solutions is introduced as a result. The main part of the paper is to show that the proposed algorithms possess an iteration complexity of $$O(1/\epsilon ^2)$$ to reach an $$\epsilon $$ -stationary solution. For prohibitively large-size tensor or machine learning models, we present a sampling-based stochastic algorithm with the same iteration complexity bound in expectation. In case the subproblems are not analytically solvable, a feasible curvilinear line-search variant of the algorithm based on retraction operators is proposed. Finally, we show specifically how the algorithms can be implemented to solve a variety of practical problems such as the NP-hard maximum bisection problem, the $$\ell _q$$ regularized sparse tensor principal component analysis and the community detection problem. Our preliminary numerical results show great potentials of the proposed methods.

This paper addresses the question of whether it can be beneficial for an optimization algorithm to follow directions of negative curvature. Although prior work has established convergence results for algorithms that integrate both descent and negative curvature steps, there has not yet been extensive numerical evidence showing that such methods offer consistent performance improvements. In this paper, we present new frameworks for combining descent and negative curvature directions: alternating two-step approaches and dynamic step approaches. The aspect that distinguishes our approaches from ones previously proposed is that they make algorithmic decisions based on (estimated) upper-bounding models of the objective function. A consequence of this aspect is that our frameworks can, in theory, employ fixed stepsizes, which makes the methods readily translatable from deterministic to stochastic settings. For deterministic problems, we show that instances of our dynamic framework yield gains in performance compared to related methods that only follow descent steps. We also show that gains can be made in a stochastic setting in cases when a standard stochastic-gradient-type method might make slow progress.

Stochastic variational inequalities (SVI) provide a unified framework for the study of a general class of nonlinear optimization and Nash-type equilibrium problems with uncertain model data. Often the true solution to an SVI cannot be found directly and must be approximated. This paper considers the use of a sample average approximation (SAA), and proposes a new method to compute confidence intervals for individual components of the true SVI solution based on the asymptotic distribution of SAA solutions. We estimate the asymptotic distribution based on one SAA solution instead of generating multiple SAA solutions, and can handle inequality constraints without requiring the strict complementarity condition in the standard nonlinear programming setting. The method in this paper uses the confidence regions to guide the selection of a single piece of a piecewise linear function that governs the asymptotic distribution of SAA solutions, and does not rely on convergence rates of the SAA solutions in probability. It also provides options to control the computation procedure and investigate effects of certain key estimates on the intervals.

In this paper, we establish the convergence of the proximal alternating direction method of multipliers (ADMM) and block coordinate descent (BCD) method for nonseparable minimization models with quadratic coupling terms. The novel convergence results presented in this paper answer several open questions that have been the subject of considerable discussion. We firstly extend the 2-block proximal ADMM to linearly constrained convex optimization with a coupled quadratic objective function, an area where theoretical understanding is currently lacking, and prove that the sequence generated by the proximal ADMM converges in point-wise manner to a primal-dual solution pair. Moreover, we apply randomly permuted ADMM (RPADMM) to nonseparable multi-block convex optimization, and prove its expected convergence for a class of nonseparable quadratic programming problems. When the linear constraint vanishes, the 2-block proximal ADMM and RPADMM reduce to the 2-block cyclic proximal BCD method and randomly permuted BCD (RPBCD). Our study provides the first iterate convergence result for 2-block cyclic proximal BCD without assuming the boundedness of the iterates. We also theoretically establish the expected iterate convergence result concerning multi-block RPBCD for convex quadratic optimization. In addition, we demonstrate that RPBCD may have a worse convergence rate than cyclic proximal BCD for 2-block convex quadratic minimization problems. Although the results on RPADMM and RPBCD are restricted to quadratic minimization models, they provide some interesting insights: (1) random permutation makes ADMM and BCD more robust for multi-block convex minimization problems; (2) cyclic BCD may outperform RPBCD for “nice” problems, and RPBCD should be applied with caution when solving general convex optimization problems especially with a few blocks.

This paper formally introduces and studies a non-cooperative multi-agent game under uncertainty. The well-known Nash equilibrium is employed as the solution concept of the game. While there are several formulations of a stochastic Nash equilibrium problem, we focus mainly on a two-stage setting of the game wherein each agent is risk-averse and solves a rival-parameterized stochastic program with quadratic recourse. In such a game, each agent takes deterministic actions in the first stage and recourse decisions in the second stage after the uncertainty is realized. Each agent's overall objective consists of a deterministic first-stage component plus a second-stage mean-risk component defined by a coherent risk measure describing the agent's risk aversion. We direct our analysis towards a broad class of quantile-based risk measures and linear-quadratic recourse functions. For this class of non-cooperative games under uncertainty, the agents' objective functions can be shown to be convex in their own decision variables, provided that the deterministic component of these functions have the same convexity property. Nevertheless, due to the non-differentiability of the recourse functions, the agents' objective functions are at best directionally differentiable. Such non-differentiability creates multiple challenges for the analysis and solution of the game, two principal ones being: (1) a stochastic multi-valued variational inequality is needed to characterize a Nash equilibrium, provided that the players' optimization problems are convex; (2) one needs to be careful in the design of algorithms that require differentiability of the objectives. Moreover, the resulting (multi-valued) variational formulation cannot be expected to be of the monotone type in general. The main contributions of this paper are as follows: (a) Prior to addressing the main problem of the paper, we summarize several approaches that have existed in the literature to deal with uncertainty in a non-cooperative game. (b) We introduce a unified formulation of the two-stage SNEP with risk-averse players and convex quadratic recourse functions and highlight the technical challenges in dealing with this game. (c) To handle the lack of smoothness, we propose smoothing schemes and regularization that lead to differentiable approximations. (d) To deal with non-monotonicity, we impose a generalized diagonal dominance condition on the players' smoothed objective functions that facilitates the application and ensures the convergence of an iterative best-response scheme. (e) To handle the expectation operator, we rely on known methods in stochastic programming that include sampling and approximation. (f) We provide convergence results for various versions of the best-response scheme, particularly for the case of private recourse functions. Overall, this paper lays the foundation for future research into the class of SNEPs that provides a constructive paradigm for modeling and solving competitive decision making problems with risk-averse players facing uncertainty; this paradigm is very much at an infancy stage of research and requires extensive treatment in order to meet its broad applications in many engineering and economics domains.

In this paper we introduce the notions of critical and noncritical multipliers for variational systems and extend to a general framework the corresponding notions by Izmailov and Solodov developed for classical Karush–Kuhn–Tucker (KKT) systems. It has been well recognized that critical multipliers are largely responsible for slow convergence of major primal–dual algorithms of optimization. The approach of this paper allows us to cover KKT systems arising in various classes of smooth and nonsmooth problems of constrained optimization including composite optimization, minimax problems, etc. Concentrating on a polyhedral subdifferential case and employing recent results of second-order subdifferential theory, we obtain complete characterizations of critical and noncritical multipliers via the problem data. It is shown that noncriticality is equivalent to a certain calmness property of a perturbed variational system and that critical multipliers can be ruled out by full stability of local minimizers in problems of composite optimization. For the latter class we establish the equivalence between noncriticality of multipliers and robust isolated calmness of the associated solution map and then derive explicit characterizations of these notions via appropriate second-order sufficient conditions. It is finally proved that the Lipschitz-like/Aubin property of solution maps yields their robust isolated calmness.

This paper considers a class of stochastic second-order-cone complementarity problems (SSOCCP), which are generalizations of the noticeable stochastic complementarity problems and can be regarded as the Karush–Kuhn–Tucker conditions of some stochastic second-order-cone programming problems. Due to the existence of random variables, the SSOCCP may not have a common solution for almost every realization . In this paper, motivated by the works on stochastic complementarity problems, we present a deterministic formulation called the expected residual minimization formulation for SSOCCP. We present an approximation method based on the Monte Carlo approximation techniques and investigate some properties related to existence of solutions of the ERM formulation. Furthermore, we experiment some practical applications, which include a stochastic natural gas transmission problem and a stochastic optimal power flow problem in radial network.

Recently, Bandeira [arXiv:1509.00824] introduced a new type of algorithm (the so-called probably certifiably correct algorithm) that combines fast solvers with the optimality certificates provided by convex relaxations. In this paper, we devise such an algorithm for the problem of k-means clustering. First, we prove that Peng and Wei's semidefinite relaxation of k-means is tight with high probability under a distribution of planted clusters called the stochastic ball model. Our proof follows from a new dual certificate for integral solutions of this semidefinite program. Next, we show how to test the optimality of a proposed k-means solution using this dual certificate in quasilinear time. Finally, we analyze a version of spectral clustering from Peng and Wei that is designed to solve k-means in the case of two clusters. In particular, we show that this quasilinear-time method typically recovers planted clusters under the stochastic ball model., Major revision from previous version. This paper is a extension of and improvement to the authors' preprint [arXiv:1505.04778]

In this paper we give an exponential lower bound for Cunningham's least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.

The nonnegative rank of a matrix A is the smallest integer r such that A can be written as the sum of r rank-one nonnegative matrices. The nonnegative rank has received a lot of attention recently due to its application in optimization, probability and communication complexity. In this paper we study a class of atomic rank functions defined on a convex cone which generalize several notions of "positive" ranks such as nonnegative rank or cp-rank (for completely positive matrices). The main contribution of the paper is a new method to obtain lower bounds for such ranks which improve on previously known bounds. Additionally the bounds we propose can be computed by semidefinite programming. The idea of the lower bound relies on an atomic norm approach where the atoms are self-scaled according to the vector (or matrix, in the case of nonnegative rank) of interest. This results in a lower bound that is invariant under scaling and that is at least as good as other existing norm-based bounds. We mainly focus our attention on the two important cases of nonnegative rank and cp-rank where our bounds satisfying interesting properties: For the nonnegative rank we show that our lower bound can be interpreted as a non-combinatorial version of the fractional rectangle cover number, while the sum-of-squares relaxation is closely related to the Lov\'asz theta number of the rectangle graph of the matrix. We also prove that the lower bound inherits many of the structural properties satisfied by the nonnegative rank such as invariance under diagonal scaling, subadditivity, etc. We also apply our method to obtain lower bounds on the cp-rank for completely positive matrices. In this case we prove that our lower bound is always greater than or equal the plain rank lower bound, and we show that it has interesting connections with combinatorial lower bounds based on edge-clique cover number.

In a recent paper, Izmailov (Math Program Ser A 147:581---590, 2014) derived an extension of Robinson's implicit function theorem for nonsmooth generalized equations in finite dimensions, which reduces to Clarke's inverse function theorem when the generalized equation is just an equation. Pales (J Math Anal Appl 209:202---220, 1997) gave earlier a generalization of Clarke's inverse function theorem to Banach spaces by employing Ioffe's strict pre-derivative. In this paper we generalize both theorems of Izmailov and Pales to nonsmooth generalized equations in Banach spaces.

This paper is devoted to the design of an efficient and convergent {semi-proximal} alternating direction method of multipliers (ADMM) for finding a solution of low to medium accuracy to convex quadratic conic programming and related problems. For this class of problems, the convergent two block semi-proximal ADMM can be employed to solve their primal form in a straightforward way. However, it is known that it is more efficient to apply the directly extended multi-block semi-proximal ADMM, though its convergence is not guaranteed, to the dual form of these problems. Naturally, one may ask the following question: can one construct a convergent multi-block semi-proximal ADMM that is more efficient than the directly extended semi-proximal ADMM? Indeed, for linear conic programming with 4-block constraints this has been shown to be achievable in a recent paper by Sun, Toh and Yang [arXiv preprint arXiv:1404.5378, (2014)]. Inspired by the aforementioned work and with the convex quadratic conic programming in mind, we propose a Schur complement based convergent semi-proximal ADMM for solving convex programming problems, with a coupling linear equality constraint, whose objective function is the sum of two proper closed convex functions plus an arbitrary number of convex quadratic or linear functions. Our convergent semi-proximal ADMM is particularly suitable for solving convex quadratic semidefinite programming (QSDP) with constraints consisting of linear equalities, a positive semidefinite cone and a simple convex polyhedral set. The efficiency of our proposed algorithm is demonstrated by numerical experiments on various examples including QSDP., 39 pages, 1 figure, 4 tables

In this paper, we analyze and discuss the well-posedness of two new variants of the so-called sweeping process, introduced by Moreau in the early 70s (Moreau in Sem Anal Convexe Montpellier, 1971) with motivation in plasticity theory. The first new variant is concerned with the perturbation of the normal cone to the moving convex subset $$C(t)$$ C ( t ) , supposed to have a bounded variation, by a Lipschitz mapping. Under some assumptions on the data, we show that the perturbed differential measure inclusion has one and only one right continuous solution with bounded variation. The second variant, for which a large analysis is made, concerns a first order sweeping process with velocity in the moving set $$C(t)$$ C ( t ) . This class of problems subsumes as a particular case, the evolution variational inequalities [widely used in applied mathematics and unilateral mechanics (Duvaut and Lions in Inequalities in mechanics and physics. Springer, Berlin, 1976]. Assuming that the moving subset $$C(t)$$ C ( t ) has a continuous variation for every $$t\in [0,T]$$ t ? [ 0 , T ] with $$C(0)$$ C ( 0 ) bounded, we show that the problem has at least a Lipschitz continuous solution. The well-posedness of this class of sweeping process is obtained under the coercivity assumption of the involved operator. We also discuss some applications of the sweeping process to the study of vector hysteresis operators in the elastoplastic model (Kreja?i in Eur J Appl Math 2:281---292, 1991), to the planning procedure in mathematical economy (Henry in J Math Anal Appl 41:179---186, 1973 and Cornet in J. Math. Anal. Appl. 96:130---147, 1983), and to nonregular electrical circuits containing nonsmooth electronic devices like diodes (Acary et al. Nonsmooth modeling and simulation for switched circuits. Lecture notes in electrical engineering. Springer, New York 2011). The theoretical results are supported by some numerical simulations to prove the efficiency of the algorithm used in the existence proof. Our methodology is based only on tools from convex analysis. Like other papers in this collection, we show in this presentation how elegant modern convex analysis was influenced by Moreau's seminal work.

Let $${S =\{x\in \mathbb{R}^n: g_1(x)\geq 0, \ldots, g_m(x)\geq 0\}}$$be a semialgebraic set defined by multivariate polynomials g i (x). Assume S is convex, compact and has nonempty interior. Let $${S_i =\{x\in \mathbb{R}^n:\, g_i(x)\geq 0\}}$$, and ∂ S (resp. ∂ S i ) be the boundary of S (resp. S i ). This paper, as does the subject of semidefinite programming (SDP), concerns linear matrix inequalities (LMIs). The set S is said to have an LMI representation if it equals the set of solutions to some LMI and it is known that some convex S may not be LMI representable (Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007). A question arising from Nesterov and Nemirovski (SIAM studies in applied mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1994), see Helton and Vinnikov in Commun Pure Appl Math 60(5):654–674, 2007 and Nemirovski in Plenary lecture, International Congress of Mathematicians (ICM), Madrid, Spain, 2006, is: given a subset S of $${\mathbb{R}^n}$$, does there exist an LMI representable set Ŝ in some higher dimensional space $${ \mathbb{R}^{n+N}}$$whose projection down onto $${\mathbb{R}^n}$$equals S. Such S is called semidefinite representable or SDP representable. This paper addresses the SDP representability problem. The following are the main contributions of this paper: (i) assume g i (x) are all concave on S. If the positive definite Lagrange Hessian condition holds, i.e., the Hessian of the Lagrange function for optimization problem of minimizing any nonzero linear function l T x on S is positive definite at the minimizer, then S is SDP representable. (ii) If each g i (x) is either sos-concave ( − ∇2 g i (x) = W(x)T W(x) for some possibly nonsquare matrix polynomial W(x)) or strictly quasi-concave on S, then S is SDP representable. (iii) If each S i is either sos-convex or poscurv-convex (S i is compact convex, whose boundary has positive curvature and is nonsingular, i.e., ∇g i (x) ≠ 0 on ∂ S i ∩ S), then S is SDP representable. This also holds for S i for which ∂ S i ∩ S extends smoothly to the boundary of a poscurv-convex set containing S. (iv) We give the complexity of Schmudgen and Putinar’s matrix Positivstellensatz, which are critical to the proofs of (i)–(iii).

This paper introduces and studies the class of differential variational inequalities (DVIs) in a finite-dimensional Euclidean space. The DVI provides a powerful modeling paradigm for many applied problems in which dynamics, inequalities, and discontinuities are present; examples of such problems include constrained time-dependent physical systems with unilateral constraints, differential Nash games, and hybrid engineering systems with variable structures. The DVI unifies several mathematical problem classes that include ordinary differential equations (ODEs) with smooth and discontinuous right-hand sides, differential algebraic equations (DAEs), dynamic complementarity systems, and evolutionary variational inequalities. Conditions are presented under which the DVI can be converted, either locally or globally, to an equivalent ODE with a Lipschitz continuous right-hand function. For DVIs that cannot be so converted, we consider their numerical resolution via an Euler time-stepping procedure, which involves the solution of a sequence of finite-dimensional variational inequalities. Borrowing results from differential inclusions (DIs) with upper semicontinuous, closed and convex valued multifunctions, we establish the convergence of such a procedure for solving initial-value DVIs. We also present a class of DVIs for which the theory of DIs is not directly applicable, and yet similar convergence can be established. Finally, we extend the method to a boundary-value DVI and provide conditions for the convergence of the method. The results in this paper pertain exclusively to systems with “index” not exceeding two and which have absolutely continuous solutions.

In this paper, we present a new framework of bi-level unconstrained minimization for development of accelerated methods in Convex Programming. These methods use approximations of the high-order proximal points, which are solutions of some auxiliary parametric optimization problems. For computing these points, we can use different methods, and, in particular, the lower-order schemes. This opens a possibility for the latter methods to overpass traditional limits of the Complexity Theory. As an example, we obtain a new second-order method with the convergence rate $$O\left( k^{-4}\right) $$ O k - 4 , where k is the iteration counter. This rate is better than the maximal possible rate of convergence for this type of methods, as applied to functions with Lipschitz continuous Hessian. We also present new methods with the exact auxiliary search procedure, which have the rate of convergence $$O\left( k^{-(3p+1)/ 2}\right) $$ O k - ( 3 p + 1 ) / 2 , where $$p \ge 1$$ p ≥ 1 is the order of the proximal operator. The auxiliary problem at each iteration of these schemes is convex.

The notion of best approximation mapping (BAM) with respect to a closed affine subspace in finite-dimensional space was introduced by Behling, Bello Cruz and Santos to show the linear convergence of the block-wise circumcentered-reflection method. The best approximation mapping possesses two critical properties of the circumcenter mapping for linear convergence. Because the iteration sequence of BAM linearly converges, the BAM is interesting in its own right. In this paper, we naturally extend the definition of BAM from closed affine subspace to nonempty closed convex set and from $\mathbb{R}^{n}$ to general Hilbert space. We discover that the convex set associated with the BAM must be the fixed point set of the BAM. Hence, the iteration sequence generated by a BAM linearly converges to the nearest fixed point of the BAM. Connections between BAMs and other mappings generating convergent iteration sequences are considered. Behling et al.\ proved that the finite composition of BAMs associated with closed affine subspaces is still a BAM in $\mathbb{R}^{n}$. We generalize their result from $\mathbb{R}^{n}$ to general Hilbert space and also construct a new constant associated with the composition of BAMs. This provides a new proof of the linear convergence of the method of alternating projections. Moreover, compositions of BAMs associated with general convex sets are investigated. In addition, we show that convex combinations of BAMs associated with affine subspaces are BAMs. Last but not least, we connect BAM with circumcenter mapping in Hilbert spaces., 31 pages and 2 figures

We address the problem of prescribing an optimal decision in a framework where the cost function depends on uncertain problem parameters that need to be learned from data. Earlier work proposed prescriptive formulations based on supervised machine learning methods. These prescriptive methods can factor in contextual information on a potentially large number of covariates to take context specific actions which are superior to any static decision. When working with noisy or corrupt data, however, such nominal prescriptive methods can be prone to adverse overfitting phenomena and fail to generalize on out-of-sample data. In this paper we combine ideas from robust optimization and the statistical bootstrap to propose novel prescriptive methods which safeguard against overfitting. We show indeed that a particular entropic robust counterpart to such nominal formulations guarantees good performance on synthetic bootstrap data. As bootstrap data is often a sensible proxy to actual out-of-sample data, our robust counterpart can be interpreted to directly encourage good out-of-sample performance. The associated robust prescriptive methods furthermore reduce to convenient tractable convex optimization problems in the context of local learning methods such as nearest neighbors and Nadaraya-Watson learning. We illustrate our data-driven decision-making framework and our novel robustness notion on a small newsvendor problem.

In this paper, we study the lower iteration complexity bounds for finding the saddle point of a strongly convex and strongly concave saddle point problem: $$\min _x\max _yF(x,y)$$ . We restrict the classes of algorithms in our investigation to be either pure first-order methods or methods using proximal mappings. For problems with gradient Lipschitz constants ( $$L_x, L_y$$ and $$L_{xy}$$ ) and strong convexity/concavity constants ( $$\mu _x$$ and $$\mu _y$$ ), the class of pure first-order algorithms with the linear span assumption is shown to have a lower iteration complexity bound of $$\Omega \,\left( \sqrt{\frac{L_x}{\mu _x}+\frac{L_{xy}^2}{\mu _x\mu _y}+\frac{L_y}{\mu _y}}\cdot \ln \left( \frac{1}{\epsilon }\right) \right) $$ , where the term $$\frac{L_{xy}^2}{\mu _x\mu _y}$$ explains how the coupling influences the iteration complexity. Under several special parameter regimes, this lower bound has been achieved by corresponding optimal algorithms. However, whether or not the bound under the general parameter regime is optimal remains open. Additionally, for the special case of bilinear coupling problems, given the availability of certain proximal operators, a lower bound of $$\Omega \left( \sqrt{\frac{L_{xy}^2}{\mu _x\mu _y}}\cdot \ln (\frac{1}{\epsilon })\right) $$ is established under the linear span assumption, and optimal algorithms have already been developed in the literature. By exploiting the orthogonal invariance technique, we extend both lower bounds to the general pure first-order algorithm class and the proximal algorithm class without the linear span assumption. As an application, we apply proper scaling to the worst-case instances, and we derive the lower bounds for the general convex concave problems with $$\mu _x = \mu _y = 0$$ . Several existing results in this case can be deduced from our results as special cases.

We develop a decomposition algorithm for distributionally-robust two-stage stochastic mixed-integer convex conic programs, and its important special case of distributionally-robust two-stage stochastic mixed-integer second order conic programs. This generalizes the algorithm proposed by Sen and Sherali [Mathematical Programming 106(2): 203–223, 2006]. We show that the proposed algorithm is finitely convergent if the second-stage problems are solved to optimality at incumbent first stage solutions, and solution to an optimization problem to identify worst-case probability distribution is available. The second stage problems can be solved using a branch and cut algorithm, or a parametric cuts based algorithm presented in this paper. The decomposition algorithm is illustrated with an example. Computational results on a stochastic programming generalization of a facility location problem show significant solution time improvements from the proposed approach. Solutions for many models that are intractable for an extensive form formulation become possible. Computational results also show that for the same amount of computational effort the optimality gaps for distributionally robust instances and their stochastic programming counterpart are similar.

Many iterative methods in applied mathematics can be thought of as fixed-point iterations, and such algorithms are usually analyzed analytically, with inequalities. In this paper, we present a geometric approach to analyzing contractive and nonexpansive fixed point iterations with a new tool called the scaled relative graph (SRG). The SRG provides a correspondence between nonlinear operators and subsets of the 2D plane. Under this framework, a geometric argument in the 2D plane becomes a rigorous proof of convergence., Published in Mathematical Programming

It is well-known that the sum of two quasiconvex functions is not quasiconvex in general, and the same occurs with the minimum. Although apparently these two statements (for the sum or minimum) have nothing in common, they are related, as we show in this paper. To develop our study, the notion of quasiconvex family is introduced, and we establish various characterizations of such a concept: one of them being the quasiconvexity of the pointwise infimum of arbitrary translations of quasiconvex functions in the family; another is the convexity of the union of any two of their sublevel sets; a third one is the quasiconvexity of the sum of the quasiconvex functions, composed with arbitrary nondecreasing functions. As a by-product, any of the aforementioned characterizations, besides providing quasiconvexity of the sum, also implies the semistrict quasiconvexity of the sum if every function in the family has the same property. Three concrete applications in quasiconvex optimization are presented: First, we establish the convexity of the (Benson) proper efficient solution set to a quasiconvex vector optimization problem; second, we derive conditions that allow us to reduce a constrained optimization problem to one with a single inequality constraint, and finally, we show a class of quasiconvex minimization problems having zero duality gap.

In this paper, a novel stochastic extra-step quasi-Newton method is developed to solve a class of nonsmooth nonconvex composite optimization problems. We assume that the gradient of the smooth part of the objective function can only be approximated by stochastic oracles. The proposed method combines general stochastic higher order steps derived from an underlying proximal type fixed-point equation with additional stochastic proximal gradient steps to guarantee convergence. Based on suitable bounds on the step sizes, we establish global convergence to stationary points in expectation and an extension of the approach using variance reduction techniques is discussed. Motivated by large-scale and big data applications, we investigate a stochastic coordinate-type quasi-Newton scheme that allows to generate cheap and tractable stochastic higher order directions. Finally, numerical results on large-scale logistic regression and deep learning problems show that our proposed algorithm compares favorably with other state-of-the-art methods.

In the Euclidean setting the proximal gradient method and its accelerated variants are a class of efficient algorithms for optimization problems with decomposable objective. In this paper, we develop a Riemannian proximal gradient method (RPG) and its accelerated variant (ARPG) for similar problems but constrained on a manifold. The global convergence of RPG is established under mild assumptions, and the O(1/k) is also derived for RPG based on the notion of retraction convexity. If assuming the objective function obeys the Rimannian Kurdyka–Łojasiewicz (KL) property, it is further shown that the sequence generated by RPG converges to a single stationary point. As in the Euclidean setting, local convergence rate can be established if the objective function satisfies the Riemannian KL property with an exponent. Moreover, we show that the restriction of a semialgebraic function onto the Stiefel manifold satisfies the Riemannian KL property, which covers for example the well-known sparse PCA problem. Numerical experiments on random and synthetic data are conducted to test the performance of the proposed RPG and ARPG.

This paper studies data-driven distributionally robust bottleneck combinatorial problems (DRBCP) with stochastic costs, where the probability distribution of the cost vector is contained in a ball of distributions centered at the empirical distribution specified by the Wasserstein distance. We study two distinct versions of DRBCP from different applications: (i) Motivated by the multi-hop wireless network application, we first study the uncertainty quantification of DRBCP (denoted by DRBCP-U), where decision-makers would like to have an accurate estimation of the worst-case value of DRBCP. The difficulty of DRBCP-U is to handle its max-min-max form. Fortunately, the alternative forms of the bottleneck combinatorial problems from their blockers allow us to derive equivalent deterministic reformulations, which can be computed via mixed-integer programs. In addition, by drawing the connection between DRBCP-U and its sampling average approximation counterpart under empirical distribution, we show that the Wasserstein radius can be chosen in the order of negative square root of sample size, improving the existing known results; and (ii) Next, motivated by the ride-sharing application, decision-makers choose the best service-and-passenger matching that minimizes the unfairness. This gives rise to the decision-making DRBCP (denoted by DRBCP-D). For DRBCP-D, we show that its optimal solution is also optimal to its sampling average approximation counterpart, and the Wasserstein radius can be chosen in a similar order as DRBCP-U. When the sample size is small, we propose to use the optimal value of DRBCP-D to construct an indifferent solution space and propose an alternative decision-robust model, which finds the best indifferent solution to minimize the empirical variance. We further show that the decision robust model can be recast as a mixed-integer program., 32 pages, 4 figures

In this paper, we consider the subcontracting and single-item lot-sizing problem with constant capacities, which is uncapacitated in subcontracting but capacitated in production. For a holistic understanding of the problem, an infinite-period model is proposed. Such a model provides a unified view of a capacitated lot-sizing problem. The usefulness of the infinite-period model is shown by the principle that the firm’s production schedule drives the subcontractor’s supply schedule. For efficient construction of the production schedule, an innovative technique–coined using the concept of the shadow period–is presented to designate the optimal marginal supply cost. The success of the infinite-period model is finally proved with the concept of the effective period, indicating the supply coverage over periods. This fills the gap between the uncapacitated techniques and the capacitated techniques, making it possible to apply the best techniques from the lot-sizing literature. With this novel approach, we improve existing algorithms and provide new ones for concave production and supply cost structures involving setups.

Quadratically constrained quadratic programs (QCQPs) are a fundamental class of optimization problems well-known to be NP-hard in general. In this paper we study conditions under which the standard semidefinite program (SDP) relaxation of a QCQP is tight. We begin by outlining a general framework for proving such sufficient conditions. Then using this framework, we show that the SDP relaxation is tight whenever the quadratic eigenvalue multiplicity, a parameter capturing the amount of symmetry present in a given problem, is large enough. We present similar sufficient conditions under which the projected epigraph of the SDP gives the convex hull of the epigraph in the original QCQP. Our results also imply new sufficient conditions for the tightness (as well as convex hull exactness) of a second order cone program relaxation of simultaneously diagonalizable QCQPs.

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019).

The paper provides a connection between Commutative Algebra and Integer Programming and contains two parts. The first one is devoted to the asymptotic behavior of integer programs with a fixed cost linear functional and the constraint sets consisting of a finite system of linear equations or inequalities with integer coefficients depending linearly on n. An integer $$N_*$$ is determined such that the optima of these integer programs are a quasi-linear function of n for all $$n\ge N_*$$ . Using results in the first part, one can bound in the second part the indices of stability of the Castelnuovo–Mumford regularities of integral closures of powers of a monomial ideal and that of symbolic powers of a square-free monomial ideal.

In this paper, we introduce some new notions of quasi efficiency and quasi proper efficiency for multiobjective optimization problems that reduce to the most important concepts of approximate and quasi efficient solutions given up to now. We establish main properties and provide characterizations for these solutions by linear and nonlinear scalarizations. With the help of quasi efficient solutions, a generalized subdifferential of a vector mapping is introduced, which generates a number of approximate subdifferentials frequently used in optimization in a unifying way. The generalized subdifferential is related to the classical subdifferential of real functions by the method of scalarization. An application of generalized subdifferential to express optimality conditions for quasi efficient solutions is also given.

We propose a theoretical framework to capture incremental solutions to cardinality constrained maximization problems. The defining characteristic of our framework is that the cardinality/support of the solution is bounded by a value $$k\in {\mathbb {N}}$$ k ∈ N that grows over time, and we allow the solution to be extended one element at a time. We investigate the best-possible competitive ratio of such an incremental solution, i.e., the worst ratio over all k between the incremental solution after k steps and an optimum solution of cardinality k. We consider a large class of problems that contains many important cardinality constrained maximization problems like maximum matching, knapsack, and packing/covering problems. We provide a general 2.618-competitive incremental algorithm for this class of problems, and we show that no algorithm can have competitive ratio below 2.18 in general. In the second part of the paper, we focus on the inherently incremental greedy algorithm that increases the objective value as much as possible in each step. This algorithm is known to be 1.58-competitive for submodular objective functions, but it has unbounded competitive ratio for the class of incremental problems mentioned above. We define a relaxed submodularity condition for the objective function, capturing problems like maximum (weighted) d-dimensional matching, maximum (weighted) (b-)matching and a variant of the maximum flow problem. We show a general bound for the competitive ratio of the greedy algorithm on the class of problems that satisfy this relaxed submodularity condition. Our bound generalizes the (tight) bound of 1.58 slightly beyond sub-modular functions and yields a tight bound of 2.313 for maximum (weighted) (b-)matching. Our bound is also tight for a more general class of functions as the relevant parameter goes to infinity. Note that our upper bounds on the competitive ratios translate to approximation ratios for the underlying cardinality constrained problems, and our bounds for the greedy algorithm carry over both.

The celebrated Motzkin–Straus formulation for the maximum clique problem provides a nontrivial characterization of the clique number of a graph in terms of the maximum value of a nonconvex quadratic function over a standard simplex. It was originally developed as a way of proving Turan’s theorem in graph theory, but was later used to develop competitive algorithms for the maximum clique problem based on continuous optimization. Clique relaxations, such as s-defective clique and s-plex, emerged as attractive, more practical alternatives to cliques in network-based cluster detection models arising in numerous applications. This paper establishes continuous cubic formulations for the maximum s-defective clique problem and the maximum s-plex problem by generalizing the Motzkin–Straus formulation to the corresponding clique relaxations. The formulations are used to extend Turan’s theorem and other known lower bounds on the clique number to the considered clique relaxations. Results of preliminary numerical experiments with the CONOPT solver demonstrate that the proposed formulations can be used to quickly compute high-quality solutions for the maximum s-defective clique problem and the maximum s-plex problem. The proposed formulations can also be used to generate interesting test instances for global optimization solvers.

In this paper, we propose a semi-metric for Markov processes that allows to bound optimal values of linear Markovian stochastic optimization problems. Similar to existing notions of distance for general stochastic processes, our distance is based on transportation metrics. As opposed to the extant literature, the proposed distance is problem specific, i.e., dependent on the data of the problem whose objective value we want to bound. As a result, we are able to consider problems with randomness in the constraints as well as in the objective function and therefore relax an assumption in the extant literature. We derive several properties of the proposed semi-metric and demonstrate its use in a stylized numerical example.

The main goal in this paper is to devise an approach to explicitly calculate the constant in the Hoffman’s error bound for (not necessarily convex) inequality systems defining convex sets. We give a constructive proof of the Hoffman’s error bound and show that we can use our method to calculate the constant at least in simple cases.

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and highly flexible manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on a large-scale rare feature selection problem.

Stochastic dual dynamic programming is a cutting plane type algorithm for multi-stage stochastic optimization originated about 30 years ago. In spite of its popularity in practice, there does not exist any analysis on the convergence rates of this method. In this paper, we first establish the number of iterations, i.e., iteration complexity, required by a basic dual dynamic programming method for solving single-scenario multi-stage optimization problems, by introducing novel mathematical tools including the saturation of search points. We then refine these basic tools and establish the iteration complexity for an explorative dual dynamic programing method proposed herein and the classic stochastic dual dynamic programming method for solving more general multi-stage stochastic optimization problems under the standard stage-wise independence assumption. Our results indicate that the complexity of these methods mildly increases with the number of stages T, in fact linearly dependent on T for discounted problems. Therefore, they are efficient for strategic decision making which involves a large number of stages, but with a relatively small number of decision variables in each stage. Without explicitly discretizing the state and action spaces, these methods might also be pertinent to the related reinforcement learning and stochastic control areas.

Utility preference robust optimization (PRO) concerns decision making problems where information on decision maker’s utility preference is incomplete and has to be elicited through partial information and the optimal decision is based on the worst case utility function elicited. A key assumption in the PRO models is that the true probability distribution is either known or can be recovered by real data generated by the true distribution. In data-driven optimization, this assumption may not be satisfied when perceived data differ from real data and consequently it raises a question as to whether statistical estimators of the PRO models based on perceived data are reliable. In this paper, we investigate the issue which is also known as qualitative robustness in the literature of statistics (Huber in Robust statistics, 3rd edn, Wiley, New York, 1981) and risk management (Kratschmer et al. in Finance Stoch 18:271–295, 2014). By utilizing the framework proposed by Kratschmer et al. (2014), we derive moderate sufficient conditions under which the optimal value and optimal solution of the PRO models are robust against perturbation of the exogenous uncertainty data, and examine how the tail behaviour of utility functions affects the robustness. Moreover, under some additional conditions on the Lipschitz continuity of the underlying functions with respect to random data, we establish quantitative robustness of the statistical estimators under the Kantorovich metric. Finally, we investigate uniform consistency of the optimal value and optimal solution of the PRO models. The results cover utility selection problems and stochastic optimization problems as special cases.

In the general equilibrium with incomplete asset markets (GEI) model, the excess demand functions are typically not continuous at the prices for which the assets have redundant returns. The reason is that, at these prices, the return matrix drops rank and households’ budget sets collapse suddenly. This discontinuity results in a serious problem for the existence and computation of general equilibrium. In this paper, we show that this problem can be resolved with a new return matrix, which has constant rank. As a function of the price vector, the continuity of this new return matrix is ensured on a subset of the price space. This enables us to handle incomplete markets using a standard homotopy path-following argument by restricting the price vector to such a subset. The proposed approach naturally provides a constructive proof for the generic existence of general equilibrium. A homotopy method can then be applied to compute equilibria in the GEI model. Numerical experiments are presented to illustrate its efficiency.

It is a challenging task to fairly compare local solvers and heuristics against each other and against global solvers. How does one weigh a faster termination time against a better quality of the found solution? In this paper, we introduce the confined primal integral, a new performance measure that rewards a balance of speed and solution quality. It emphasizes the early part of the solution process by using an exponential decay. Thereby, it avoids that the order of solvers can be inverted by choosing an arbitrarily large time limit. We provide a closed analytic formula to compute the confined primal integral a posteriori and an incremental update formula to compute it during the run of an algorithm. For the latter, we show that we can drop one of the main assumptions of the primal integral, namely the knowledge of a fixed reference solution to compare against. Furthermore, we prove that the confined primal integral is a transitive measure when comparing local solves with different final solution values. Finally, we present a computational experiment where we compare a local MINLP solver that uses certain classes of cutting planes against a solver that does not. Both versions show very different tendencies w.r.t. average running time and solution quality, and we use the confined primal integral to argue which of the two is the preferred setting.

This paper develops new extremal principles of variational analysis that are motivated by applications to constrained problems of stochastic programming and semi-infinite programming without smoothness and/or convexity assumptions. These extremal principles concern measurable set-valued mappings/multifunctions with values in finite-dimensional spaces and are established in both approximate and exact forms. The obtained principles are instrumental to derive via variational approaches integral representations and upper estimates of regular and limiting normals cones to essential intersections of sets defined by measurable multifunctions, which are in turn crucial for novel applications to stochastic and semi-infinite programming., Comment: 26 pages

In this paper, we devise new relaxations for composite functions, which improve the prevalent factorable relaxations, without introducing additional variables, by exploiting the inner-function structure. We outer-approximate inner-functions using arbitrary under- and over-estimators and then convexify the outer-function over a polytope P, which models the ordering relationships between the inner-functions and their estimators and utilizes bound information on the inner-functions as well as on the estimators. We show that there is a subset Q of P, with significantly simpler combinatorial structure, such that the separation problem of the graph of the outer-function over P is polynomially equivalent, via a fast combinatorial algorithm, to that of its graph over Q. We specialize our study to consider the product of two inner-functions with one non-trivial underestimator for each inner-function. For the corresponding polytope P, we show that there are eight valid inequalities besides the four McCormick inequalities, which improve the factorable relaxation. Finally, we show that our results generalize to simultaneous convexification of a vector of outer-functions.

The Steklov function $$\mu _f(\cdot ,t)$$ is defined to average a continuous function f at each point of its domain by using a window of size given by $$t>0$$ . It has traditionally been used to approximate f smoothly with small values of t. In this paper, we first find a concise and useful expression for $$\mu _f$$ for the case when f is a multivariate quartic polynomial. Then we show that, for large enough t, $$\mu _f(\cdot ,t)$$ is convex; in other words, $$\mu _f(\cdot ,t)$$ convexifies f. We provide an easy-to-compute formula for t with which $$\mu _f$$ convexifies certain classes of polynomials. We present an algorithm which constructs, via an ODE involving $$\mu _f$$ , a trajectory x(t) emanating from the minimizer of the convexified f and ending at x(0), an estimate of the global minimizer of f. For a family of quartic polynomials, we provide an estimate for the size of a ball that contains all its global minimizers. Finally, we illustrate the working of our method by means of numerous computational examples.

The paper is devoted to the existence of global optimal solutions for a general class of nonsmooth problems of constrained vector optimization without boundedness assumptions on constraint set. The main attention is paid to the two major notions of optimality in vector problems: Pareto efficiency and proper efficiency in the sense of Geoffrion. Employing adequate tools of variational analysis and generalized differentiation, we first establish relationships between the notions of properness, M-tameness, and the Palais–Smale conditions formulated for the restriction of the vector cost mapping on the constraint set. These results are instrumental to derive verifiable necessary and sufficient conditions for the existence of Pareto efficient solutions in vector optimization. Furthermore, the developed approach allows us to obtain new sufficient conditions for the existence of Geoffrion-properly efficient solutions to such constrained vector problems.