In this paper, we present a link between preference-based and multiobjective sequential decision-making. While transforming a multiobjective problem to a preference-based one is quite natural, the other direction is a bit less obvious. We present how this transformation (from preference-based to multiobjective) can be done under the classic condition that preferences over histories can be represented by additively decomposable utilities and that the decision criterion to evaluate policies in a state is based on expectation. This link yields a new source of multiobjective sequential decision-making problems (i.e., when reward values are unknown) and justifies the use of solving methods developed in one setting in the other one.
Still another approach to solve multiobjective problems is to assume the existence of a scalarizing function @math , which, given a vector @math , returns a scalar valuation. Two cases can be considered: @math can be either linear @cite_23 or nonlinear @cite_32 @cite_31 @cite_29 .
@cite_31 @cite_29 @cite_32 @cite_23
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In this paper, we study the mixed linear regression (MLR) problem, where the goal is to recover multiple underlying linear models from their unlabeled linear measurements. We propose a non-convex objective function which we show is locally strongly convex in the neighborhood of the ground truth. We use a tensor method for initialization so that the initial models are in the local strong convexity region. We then employ general convex optimization algorithms to minimize the objective function. To the best of our knowledge, our approach provides first exact recovery guarantees for the MLR problem with @math components. Moreover, our method has near-optimal computational complexity @math as well as near-optimal sample complexity @math for constant @math . Furthermore, we show that our non-convex formulation can be extended to solving the subspace clustering problem as well. In particular, when initialized within a small constant distance to the true subspaces, our method converges to the global optima (and recovers true subspaces) in time linear in the number of points. Furthermore, our empirical results indicate that even with random initialization, our approach converges to the global optima in linear time, providing speed-up of up to two orders of magnitude. One way of solving multiple objective mathematical programming problems is finding discrete representations of the efficient set. A modified goal of finding good discrete representations of the efficient set would contribute to the practicality of vector maximization algorithms. We define coverage, uniformity and cardinality as the three attributes of quality of discrete representations and introduce a framework that includes these attributes in which discrete representations can be evaluated, compared to each other, and judged satisfactory or unsatisfactory by a Decision Maker. We provide simple mathematical programming formulations that can be used to compute the coverage error of a given discrete representation. Our formulations are practically implementable when the problem under study is a multiobjective linear programming problem. We believe that the interactive algorithms along with the vector maximization methods can make use of our framework and its tools. We propose an approximate Newton method for solving the coupled nonlinear system @math and @math where @math , @math , @math and @math . The method involves applying the basic iteration S of a general solver for the equation @math , withtfixed. It is therefore well suited for problems for which such a solver already exists or can be implemented more efficiently than a solver for the coupled system. We derive conditions for S under which the method is locally convergent. Basically, if S is sufficiently contractive for G, then convergence for the coupled system is guaranteed. Otherwise, we show how to construct an @math from S for which convergence is assured. These results are applied to continuation methods where N represents a pseudo-arclength condition. We show that under certain conditions the algorithm converges if S is convergent for G. Numerical results are given for a two-level nonlinear multi-grid solver applied ... In this paper, we consider conic programming problems whose constraints consist of linear equalities, linear inequalities, a nonpolyhedral cone, and a polyhedral cone. A convenient way for solving this class of problems is to apply the directly extended alternating direction method of multipliers (ADMM) to its dual problem, which has been observed to perform well in numerical computations but may diverge in theory. Ideally, one should find a convergent variant which is at least as efficient as the directly extended ADMM in practice. We achieve this goal by designing a convergent semiproximal ADMM (called sPADMM3c for convenience) for convex programming problems having three separable blocks in the objective function with the third part being linear. At each iteration, the proposed sPADMM3c takes one special block coordinate descent (BCD) cycle with the order @math , instead of the usual @math Gauss--Seidel BCD cycle used in the nonconvergent directly extended 3-block ADMM, for updating the variable blocks. Our numerical experiments demonstrate that the convergent method is at least 20 faster than the directly extended ADMM with unit step-length for the vast majority of about 550 large-scale doubly nonnegative semidefinite programming problems with linear equality and or inequality constraints. This confirms that at least for conic convex programming, one can design a convergent and efficient ADMM with a special BCD cycle of updating the variable blocks. We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order- @math tensor @math of the form @math , where @math is a signal-to-noise ratio, @math is a unit vector, and @math is a random noise tensor, the goal is to recover the planted vector @math . For the case that @math has iid standard Gaussian entries, we give an efficient algorithm to recover @math whenever @math , and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of @math . The previous best algorithms with provable guarantees required @math . In the regime @math , natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree- @math sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree- @math sum-of-squares relaxations break down for @math , which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings. In a recent paper, Hou and Shi introduced a new adaptive data analysis method to analyze nonlinear and non-stationary data. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions of the form @math , where @math , @math consists of the functions smoother than @math and @math . This problem was formulated as a nonlinear @math optimization problem and an iterative nonlinear matching pursuit method was proposed to solve this nonlinear optimization problem. In this paper, we prove the convergence of this nonlinear matching pursuit method under some sparsity assumption on the signal. We consider both well-resolved and sparse sampled signals. In the case without noise, we prove that our method gives exact recovery of the original signal. We consider the problem of handling binary constraints in optimization problems. We review methods for solving binary quadratic programs (BQPs), such as the spectral method and semidefinite programming relaxations. We then discuss two new methods for handling these constraints. The first involves the introduction of an extra unconstrained variable along with the alternating direction method of multipliers (ADMM), which has now also appeared in other recent literature. This allows the effect of the binary variables to be decoupled when they are minimized over. The second involves rewarding the one-norm while restricting the infinity-norm, based on a reformulation of the original problem. The piecewise linearity of the negative penalty results in the problem being convex until it hits a critical point, at which point the parameters of this linear term can be changed. These two methods can be applied to any problems which are convex except for binary constraints. In addition to testing them on BQPs, we show the efficacy of these approaches on point segmentation and image segmentation problems. We study identifiability for bilinear inverse problems under sparsity and subspace constraints. We show that, up to a global scaling ambiguity, almost all such maps are injective on the set of pairs of sparse vectors if the number of measurements @math exceeds @math , where @math and @math denote the sparsity of the two input vectors, and injective on the set of pairs of vectors lying in known subspaces of dimensions @math and @math if @math . We also prove that both these bounds are tight in the sense that one cannot have injectivity for a smaller number of measurements. Our proof technique draws from algebraic geometry. As an application we derive optimal identifiability conditions for the deconvolution problem, thus improving on recent work of [1]. This paper considers recovering @math -dimensional vectors @math , and @math from their circular convolutions @math . The vector @math is assumed to be @math -sparse in a known basis that is spread out in the Fourier domain, and each input @math is a member of a known @math -dimensional random subspace. We prove that whenever @math , the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey @math . By “diverse inputs,” we mean that the @math ’s belong to different, generic subspaces. To the best of our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which @math belongs is not fixed but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients and the delays in the channel impulse response @math are unknown. The encoder codes the @math -dimensional message vectors randomly and then transmits coded messages @math ’s over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than @math , and the number of messages is roughly at least @math . An important issue in multiobjective optimization is the quantitative comparison of the performance of different algorithms. In the case of multiobjective evolutionary algorithms, the outcome is usually an approximation of the Pareto-optimal set, which is denoted as an approximation set, and therefore the question arises of how to evaluate the quality of approximation sets. Most popular are methods that assign each approximation set a vector of real numbers that reflect different aspects of the quality. Sometimes, pairs of approximation sets are also considered. In this study, we provide a rigorous analysis of the limitations underlying this type of quality assessment. To this end, a mathematical framework is developed which allows one to classify and discuss existing techniques. Many conjugate gradient-like methods for solving linear systems @math use recursion formulas for updating residual vectors instead of computing the residuals directly. For such methods it is shown that the difference between the actual residuals and the updated approximate residual vectors generated in finite precision arithmetic depends on the machine precision @math and on the maximum norm of an iterate divided by the norm of the true solution. It is often observed numerically, and can sometimes be proved, that the norms of the updated approximate residual vectors converge to zero or, at least, become orders of magnitude smaller than the machine precision. In such cases, the actual residual norm reaches the level @math times the maximum ratio of the norm of an iterate to that of the true solution. Using exact arithmetic theory to bound the size of the iterates, we give a priori estimates of the size of the final residual for a number of algorithms. How ro initialize an algorithm to solve an optimization problem is of great theoretical and practical importance. In the simplex method for linear programming this issue is resolved by either the two-phase approach or using the so-called big M technique. In the interior point method, there is a more elegant way to deal with the initialization problem. Viz. the self-dual embedding technique proposed by Ye, Todd and Mizuno [30]. For linear programming this technique makes it possible to identify an optimal solution or conclude the problem to be infeasible unbounded by solving tts embedded self-dual problem The embedded self-dual problem has a trivial initial solution and has the same stmctare as the original problem. Hence. it eliminates the need to consider the initialization problem at all. In this paper, we extend this approach to solve general conic convex programming, including semidefinite programmng. Since a nonlinear conic convex programming problem may lack the so-called strict complementarity prop... We consider linear problems in fields, ordered fields, discretely valued fields (with finite residue field or residue field of characteristic zero) and fields with finitely many independent orderings and discrete valuations. Most of the fields considered will be of characteristic zero. Formally, linear statements about these structures (with parameters) are given by formulas of the respective first-order language, in which all bound variables occur only linearly. We study symbolic algorithms (linear elimination procedures) that reduce linear formulas to linear formulas of a very simple form, i.e. quantifier-free linear formulas, and algorithms (linear decision procedures) that decide whether a given linear sentence holds in all structures of the given class. For all classes of fields considered, we find linear elimination procedures that run in double exponential space and time. As a consequence, we can show that for fields (with one or several discrete valuations), linear statements can be transferred from characteristic zero to prime characteristic p, provided p is double exponential in the length of the statement. (For similar bounds in the non-linear case, see Brown, 1978.) We find corresponding linear decision procedures in the Berman complexity classes @[email protected]?NSTA(*,2^c^n,dn) for d = 1, 2. In particular, all hese procedures run in exponential space. The technique employed is quantifier elimination via Skolem terms based on Ferrante & Rackoff (1975). Using ideas of Fischer & Rabin (1974), Berman (1977), Furer (1982), we establish lower bounds for these problems showing that our upper bounds are essentially tight. For linear formulas with a bounded number of quantifiers all our algorithms run in polynomial time. For linear formulas of bounded quantifier alternation most of the algorithms run in time 2^O^(^n^^^k^) for fixed k. An algorithm for solving the problem: minimize @math (a convex function) subject to @math , @math , each @math a concave function, is presented. Specifically, the function [ P [ x,t,r_k ] f( x ) + r_k^ - 1 [ g_i ( x ) - t_i ] ^2 ] is minimized over all x, nonnegative t, for a strictly decreasing null sequence @math . This extends the work of T. Pietrzykowski [5]. It is proved that for every @math , there exists a finite point @math which minimizes P, and which solves the convex programming problem as @math . This algorithm is similar to the Sequential Unconstrained Minimization Technique (SUMT) [1] in that it solves the (Wolfe) dual programming problem [6]. It differs from SUMT in that (1) it approaches the optimum from the region of infeasibility (i.e., it is a relaxation technique), (2) it does not require a nonempty interior to the nonlinearly constrained region, (3) no separate feasibilit... First-order methods have been popularly used for solving large-scale problems. However, many existing works only consider unconstrained problems or those with simple constraint. In this paper, we develop two first-order methods for constrained convex programs, for which the constraint set is represented by affine equations and smooth nonlinear inequalities. Both methods are based on the classic augmented Lagrangian function. They update the multipliers in the same way as the augmented Lagrangian method (ALM) but employ different primal variable updates. The first method, at each iteration, performs a single proximal gradient step to the primal variable, and the second method is a block update version of the first one. For the first method, we establish its global iterate convergence as well as global sublinear and local linear convergence, and for the second method, we show a global sublinear convergence result in expectation. Numerical experiments are carried out on the basis pursuit denoising and a convex quadratically constrained quadratic program to show the empirical performance of the proposed methods. Their numerical behaviors closely match the established theoretical results. We consider the problem of minimizing the sum of a smooth function @math with a bounded Hessian and a nonsmooth function. We assume that the latter function is a composition of a proper closed function @math and a surjective linear map @math , with the proximal mappings of @math , @math , simple to compute. This problem is nonconvex in general and encompasses many important applications in engineering and machine learning. In this paper, we examined two types of splitting methods for solving this nonconvex optimization problem: the alternating direction method of multipliers and the proximal gradient algorithm. For the direct adaptation of the alternating direction method of multipliers, we show that if the penalty parameter is chosen sufficiently large and the sequence generated has a cluster point, then it gives a stationary point of the nonconvex problem. We also establish convergence of the whole sequence under an additional assumption that the functions @math and @math are semialgebraic. Further... We discuss problems in multiobjective optimization, in which solutions to a combinatorial optimization problem are evaluated with respect to several cost criteria, and we are interested in the trade-off between these objectives, the so-called Pareto curve. The Pareto curve has typically an exponential number of points. However, it turns out that, under general conditions, there is a polynomially succinct curve that approximates the Pareto curve within any desired accuracy. The central computational question is whether such an approximate curve can be constructed efficiently (in polynomial time). We discuss conditions under which this is the case. We examine in more detail the class of linear multiobjective problems, and relate the multiobjective approximation to the single objective case. We will discuss also problems in multiobjective query optimization. List of Examples Preface 1. Introduction. Boundary Value Problems for Ordinary Differential Equations Boundary Value Problems in Applications 2. Review of Numerical Analysis and Mathematical Background. Errors in Computation Numerical Linear Algebra Nonlinear Equations Polynomial Interpolation Piecewise Polynomials, or Splines Numerical Quadrature Initial Value Ordinary Differential Equations Differential Operators and Their Discretizations 3. Theory of Ordinary Differential Equations. Existence and Uniqueness Results Green's Functions Stability of Initial Value Problems Conditioning of Boundary Value Problems 4. Initial Value Methods. Introduction. Shooting Superposition and Reduced Superposition Multiple Shooting for Linear Problems Marching Techniques for Multiple Shooting The Riccati Method Nonlinear Problems 5. Finite Difference Methods. Introduction Consistency, Stability, and Convergence Higher-Order One-Step Schemes Collocation Theory Acceleration Techniques Higher-Order ODEs Finite Element Methods 6. Decoupling. Decomposition of Vectors Decoupling of the ODE Decoupling of One-Step Recursions Practical Aspects of Consistency Closure and Its Implications 7. Solving Linear Equations. General Staircase Matrices and Condensation Algorithms for the Separated BC Case Stability for Block Methods Decomposition in the Nonseparated BC Case Solution in More General Cases 8. Solving Nonlinear Equations. Improving the Local Convergence of Newton's Method Reducing the Cost of the Newton Iteration Finding a Good Initial Guess Further Remarks on Discrete Nonlinear BVPS 9. Mesh Selection. Introduction Direct Methods A Mesh Strategy for Collocation Transformation Methods General Considerations 10. Singular Perturbations. Analytical Approaches Numerical Approaches Difference Methods Initial Value Methods 11. Special Topics. Reformulation of Problems in 'Standard' Form Generalized ODEs and Differential Algebraic Equations Eigenvalue Problems BVPs with Singularities Infinite Intervals Path Following, Singular Points and Bifurcation Highly Oscillatory Solutions Functional Differential Equations Method of Lines for PDEs Multipoint Problems On Code Design and Comparison Appendix A. A Multiple Shooting Code Appendix B. A Collocation Code References Bibliography Index. We consider the adversarial convex bandit problem and we build the first @math -time algorithm with @math -regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves @math -regret, and we show that a simple variant of this algorithm can be run in @math -time per step at the cost of an additional @math factor in the regret. These results improve upon the @math -regret and @math -time result of the first two authors, and the @math -regret and @math -time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve @math -regret, and moreover that this regret is unimprovable (the current best lower bound being @math and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order @math . Consider a network of @math n processes, where each process inputs a @math d-dimensional vector of reals. All processes can communicate directly with others via reliable FIFO channels. We discuss two problems. The multidimensional Byzantine consensus problem, for synchronous systems, requires processes to decide on a single @math d-dimensional vector @math v?Rd, inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. Also, the multidimensional Byzantine approximate agreement (MBAA) problem, for asynchronous systems, requires processes to decide on multiple @math d-dimensional vectors in @math Rd, all within a fixed Euclidean distance @math ∈ of each other, and inside the convex hull of @math d-dimensional vectors that were input by the non-faulty processes. We obtain the following results for the problems above, while tolerating up to @math f Byzantine failures in systems with complete communication graphs: (1) In synchronous systems, @math n>max 3f,(d+1)f is necessary and sufficient to solve the multidimensional consensus problem. (2) In asynchronous systems, @math n>(d+2)f is necessary and sufficient to solve the multidimensional approximate agreement problem. Our sufficiency proofs are constructive, giving explicit protocols for the problems. In particular, for the MBAA problem, we give two protocols with strictly different properties and applications.