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In this paper, we characterize the monoid of endomorphisms of the semigroup of all monotone full transformations of a finite chain, as well as the monoids of endomorphisms of the semigroup of all monotone partial transformations and of the semigroup of all monotone partial permutations of a finite chain. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the existence of a positive solution for a class of nonvariational fractional elliptic system with concave and convex nonlinearities in two cases. The paper is divided in two parts: In the first one, for general nonlinearity with subcritical or critical growth, we use Galerkin's method and an approximation argument to show the existence of a solution for the system considered. In the second part, for special cases (which include the power case), we remove the restriction on the growth of the nonlinearity and use sub-supersolution, monotone iteration and a comparison argument to obtain a solution for the system considered. [ABSTRACT FROM AUTHOR]

In this article, we introduced the idea of lacunary weak convergence of sequences defined by the class of Orlicz function. Various topological and algebraic properties, such as solidity, symmetry, and monotonicity, have been studied. To validate the failure instances, we gave appropriate examples and detailed discussions. Additionally, we have established some inclusion relationships between these spaces. [ABSTRACT FROM AUTHOR]

In the framework of p-uniformly convex uniformly and smooth Banach spaces, we introduce the split common generalized variational incusion problem with multiple output sets. Using a Halpern iterative method, we propose a strong convergence algorithm for approximating a common solution of this problem and fixed point of a Bregman firmly nonexpansive mapping. Furthermore, we give a theoretical application to split common minimization problem with multiple output sets. We also report some numerical examples to illustrate the performance of our proposed method. [ABSTRACT FROM AUTHOR]

The article reveals concept of lpN(X), 1 ≤ p < ∞, the p – absolutely summable neutrosophic valued sequence space. We have proved that it is a neutrosophic normed linear space valued sequence space. We have studied its different algebraic and topological properties. We have also established some inclusion results. The main aim is to introduce the notion of neutrosophic normed linear space lpN(X), 1 ≤ p < ∞, and to show that it is a complete neutrosophic sequence space. [ABSTRACT FROM AUTHOR]

Beta distributions are commonly used to model proportion valued response variables, often encountered in longitudinal studies. In this article, we develop semi-parametric Beta regression models for proportion valued responses, where the aggregate covariate effect is summarized and flexibly modeled, using a interpretable monotone time-varying single index transform of a linear combination of the potential covariates. We utilize the potential of single index models, which are effective dimension reduction tools and accommodate link function misspecification in generalized linear mixed models. Our Bayesian methodology incorporates the missing-at-random feature of the proportion response and utilize Hamiltonian Monte Carlo sampling to conduct inference. We explore finite-sample frequentist properties of our estimates and assess the robustness via detailed simulation studies. Finally, we illustrate our methodology via application to a motivating longitudinal dataset on obesity research recording proportion body fat. [ABSTRACT FROM AUTHOR]

In this work, we introduce an inertial proximal algorithm for solving multivalued variational inequality problems in a real Hilbert space. By using self-adaptive and inertial techniques via proximal operators, we establish the weak convergence of the iteration sequences generated by these algorithms when the multivalued cost mappings associated with the problems are monotone and Lipschitz continuous. Moreover, we present the nonasymptotic $ O(\frac {1}{k}) $ O (1 k) convergence rate of the proposed algorithms. We also provide some numerical examples to illustrate the accuracy and efficiency of our algorithms by comparing with other recent algorithms in the literature. [ABSTRACT FROM AUTHOR]

This paper addresses L-fuzzy topologies induced by L-G-filters and studies the categorical relations between L-G-filter spaces and L-fuzzy topological spaces. Three functors from the category of L-G-filter spaces to the category of Lfuzzy topological spaces are obtained. Having introduced the concept of monotone L-fuzzy topologies, the study inquires into the sum, subspace, product, quotient and the lattice structure of such topologies. [ABSTRACT FROM AUTHOR]

We prove a general homogenization result for monotone parabolic problems with an arbitrary number of microscopic scales in space as well as in time, where the scale functions are not necessarily powers of the scale parameter ε. The main tools for the homogenization procedure are multiscale convergence and very weak multiscale convergence, both adapted to evolution problems. [ABSTRACT FROM AUTHOR]

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to study in which posets some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again.

Let h : A → A and be a partial order on A. We deal with properties of oriented graphs which correspond to the algebra (A , h) in the case that h is monotone with respect to . We derive that every mono-unary algebra except connected one with a cycle of odd length has the property that there exists a nontrivial partial order such that (A , h) is monotone with respect to it. All mono-unary algebras such that there exists a linear order such that h is monotone with respect to this order will be described; if the number of components of (A , h) is infinite, then the number of such orders is equal to the cardinality of the power set of A. [ABSTRACT FROM AUTHOR]

In this work, we analyze some convergent properties of a projection and contraction algorithm for solving a variational inequality problem, where the feasible domain is the solution set of an affine variational inequality problem. We prove that, for solving the problem where the second cost mapping is affine and not necessary for monotone properties, any iterative sequence generated by the algorithm converges to a unique solution provided that the first cost mapping is strongly monotone and Lipschitz continuous. Computational errors of the algorithm are showed. Finally, some preliminary numerical experiences and comparisons are also reported. [ABSTRACT FROM AUTHOR]

We use strongly pseudocontractions to regularise a class of accretive variational inequalities in more general settings, the solutions are sought in the set of fixed points of another pseudocontraction. In this paper, we consider an implicit scheme that can be used to find a solution of a class of accretive variational inequalities. Our results improved and generalise some results of Yaqin and Chen.

In this paper, we define a new area-type filter algorithm based on the trust-region method. A relaxed trust-region quadratic correction subproblem is proposed to compute the trial direction at the current point. Consider the objective function and the constraint violation function at the current point as a point pair. We divide the point pairs into different partitions by the dominant region of the filter and calculate the contributions of the point pairs to the area of the filter separately. Different from the conventional filter, we define the contribution as the filter acceptance criterion for the trial point. The nonmonotone area-average form is also adopted in the filter mechanism. In this paper, monotone and nonmonotone methods are proposed and compared with the numerical values. Furthermore, the algorithm is proved to be convergent under some reasonable assumptions. The numerical experiment shows the effectiveness of the algorithm.

We introduce an integral representation of the Monge--Ampère equation, which leads to a new finite difference method based upon numerical quadrature. The resulting scheme is monotone and fits immediately into existing convergence proofs for the Monge--Ampère equation with either Dirichlet or optimal transport boundary conditions. The use of higher-order quadrature schemes allows for substantial reduction in the component of the error that depends on the angular resolution of the finite difference stencil. This, in turn, allows for significant improvements in both stencil width and formal truncation error. The resulting schemes can achieve a formal accuracy that is arbitrarily close to O (h2), which is the optimal consistency order for monotone approximations of second-order operators. We present three different implementations of this method. The first two exploit the spectral accuracy of the trapezoid rule on uniform angular discretizations to allow for computation on a nearest-neighbors finite difference stencil over a large range of grid refinements. The third uses higher-order quadrature to produce superlinear convergence while simultaneously utilizing narrower stencils than other monotone methods. Computational results are presented in two dimensions for problems of various regularity. [ABSTRACT FROM AUTHOR]

We define several sorts of mappings on a poset like monotone, strictly monotone, upper cone preserving and variants of these. Our aim is to study in which posets some of these mappings coincide. We define special mappings determined by two elements and investigate when these are strictly monotone or upper cone preserving. If the considered poset is a semilattice then its monotone mappings coincide with semilattice homomorphisms if and only if the poset is a chain. Similarly, we study posets which need not be semilattices but whose upper cones have a minimal element. We extend this investigation to posets that are direct products of chains or an ordinal sum of an antichain and a finite chain. We characterize equivalence relations induced by strongly monotone mappings and show that the quotient set of a poset by such an equivalence relation is a poset again. [ABSTRACT FROM AUTHOR]

The present paper deals with penalty function method for K -constrained variational inequality problem on Hadamard manifolds. It transforms the constrained problem into an unconstrained problem and named it a penalized variational inequality problem. We establish sufficient condition under the assumption of coercivity for solving both problems and illustrate by a non-trivial example. It is shown that the sequence of a solution to the penalized variational inequality problem has at least one limit point within the feasible region. Moreover, it is also observed that any limit point of the sequence of a solution to the penalized variational inequality problem also is a solution to the original problem. [ABSTRACT FROM AUTHOR]

The tensor complementarity problem is a special instance in the class of nonlinear complementarity problems, which has many applications in multi-person noncooperative games, hypergraph clustering problems and traffic equilibrium problems. Two most important research issues are how to identify the solvability and how to solve such a problem via analyzing the structure of the involved tensor. In this paper, based on the concept of monotone mappings, we introduce a new class of structured tensors and the corresponding monotone tensor complementarity problem. We show that the solution set of the monotone tensor complementarity problem is nonempty and compact under the feasibility assumption. Moreover, a necessary and sufficient condition for ensuring the feasibility is given via analyzing the structure of the involved tensor. Based on the Huber function, we propose a regularized smoothing Newton method to solve the monotone tensor complementarity problem and establish its global convergence. Under some mild assumptions, we show that the proposed algorithm is superlinearly convergent. Preliminary numerical results indicate that the proposed algorithm is very promising. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce an inertial Halpern-type iterative algorithm for approximating a zero of the sum of two monotone operators in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. Strong convergence of the sequence generated by our proposed algorithm is established by means of some new geometric inequalities proved in this paper that are of independent interest. Furthermore, numerical simulations in image restoration and compressed sensing problems are also presented. Finally, the performance of the proposed method is compared with that of some existing methods in the literature.

In this paper, we propose a concept of a lattice pseudo-submodular (LPS) function and consider maximizing a monotone continuous real LPS function ϕ(x) under a convex polytope constraint. The concept of LPS function was proposed to describe the properties of some discrete functions or nonconvex continuous functions. It is a generalization of the lattice submodular function. For the real LPS maximization problem, we design the monotone Pseudo Frank-Wolfe (PFW) algorithm by taking advantage of the second derivative bound. The PFW algorithm iterates by constantly optimize linear gradient function vT∇ϕ(x), and finally outputs the solution. We theoretically prove that PFK algorithm has an approximation ratio of (1 − e−( β β+1 )2) (where β ∈ (0, 1]), and it needs at least O(nζ) rounds (where ζ is a parameter given in advance). The PFW algorithm is also useful for multilinear extension of discrete lattice pseudo-submodular maximization problems. [ABSTRACT FROM AUTHOR]

Personalized medicine allows individuals to choose the best fit of their treatments based on their characteristics through an individualized treatment regime. In this paper, we develop a pool adjacent violators algorithm–assisted learning method to find the optimal individualized treatment regime under the monotone single‐index outcome gain model. The proposed estimator is more efficient than peers, and it is robust to the misspecification of the propensity score model or the baseline regression model. The optimal treatment regime is also robust to the misspecification of the functional form of the expected outcome gain model. Simulation studies verified our theoretical results. We also provide an estimate of the expected outcome gain model. Plotting the expected outcome gain versus an individual's characteristics index can visualize how significant the treatment effect is over the control. We apply the proposed method to an AIDS study. [ABSTRACT FROM AUTHOR]

This paper is devoted to investigating the existence of solution to a class of reflected forward-backward stochastic differential equations driven by G-Brownian motion (G-RFBSDEs). We construct a solution to the equations by monotone convergence argument when the generator of backward equation and the drift of forward equation satisfy the monotonicity and uniformly continuous condition. [ABSTRACT FROM AUTHOR]

We study topologically monotone surjective W 1 , n -maps of finite distortion f : Ω → Ω ′ , where Ω , Ω ′ are domains in R n , n ≥ 2 . If the outer distortion function K f ∈ L loc p (Ω) with p ≥ n - 1 , then any such map f is known to be homeomorphic, and hence the fibers f - 1 { y } are singletons. We show that as the exponent of integrability p of the distortion function K f increases in the range 1 / (n - 1) ≤ p < n - 1 , then for increasingly many k ∈ { 0 , ⋯ , n } depending on p, the k:th rational homology group H k (f - 1 { y } ; Q) of any reasonably tame fiber f - 1 { y } of f is equal to that of a point. In particular, if p ≥ (n - 2) / 2 then this is true for all k ∈ { 0 , ⋯ , n } . We also formulate a Sobolev realization of a topological example by Bing of a monotone f : R 3 → R 3 with homologically non-trivial fibers. This example has K f ∈ L loc 1 / 2 - ε (R 3) for all ε > 0 , which shows that our result is sharp in the case n = 3 . [ABSTRACT FROM AUTHOR]

In this work, an inertial Halpern-type algorithm involving monotone operators is proposed in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. Strong convergence of the iterates generated by the algorithm is proved to a zero of sum of two monotone operators. Furthermore, an application of the method to image recovery problems is presented. In addition, a numerical example on the classical Banach space l 3 2 (R) is presented to support the main theorem. Finally, the performance of the proposed algorithm is compared with that of some existing algorithms in the literature. [ABSTRACT FROM AUTHOR]

Abstract In this paper, we introduce two subgradient extragradient-type algorithms for solving variational inequality problems in the real Hilbert space. The first one can be applied when the mapping f is strongly pseudomonotone (not monotone) and Lipschitz continuous. The first algorithm only needs two projections, where the first projection onto closed convex set C and the second projection onto a half-space C k $C_{k}$ . The strong convergence theorem is also established. The second algorithm is relaxed and self-adaptive; that is, at each iteration, calculating two projections onto some half-spaces and the step size can be selected in some adaptive ways. Under the assumption that f is monotone and Lipschitz continuous, a weak convergence theorem is provided. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms.

Flux limiters are widely used in numerical simulations to prevent spurious oscillation in the flow with strong property gradients. However, applying flux limiter on flow without strong property gradient such as advection-diffusion flow can cause errors. This article discusses the errors caused by several flux limiters in advection-diffusion flow solution. A method for applying one-dimensional limiters to two-dimensional unstructured mesh was also suggested. The error was measured by comparing the finite volume solution of a test case with a reference solution. The study shows that the calculation error of second-order finite volume with flux limiter was higher than that of second-order finite volume without limiter. However, the error of third-order finite volume with flux limiter is less than that of second-order without flux limiter. Among the flux limiters tested in this study, Venkatakrishnan’s flux limiter produces the highest error, followed by Van leer’s limiter, EULER and SMART limiter.

This paper presents a study of the influence of plastic and non-plastic fines on the resistance of sands to liquefaction. The study is carried out with the aid of a triaxial apparatus on reconstituted sand-fines mixtures at low fines content (≤ 5%). A description of the soils used in this study is first presented followed by the presentation of results of undrained monotonic liquefaction tests performed on sand specimens containing different fines fractions. Experimental results show that the increase in non-plastic fines content up to 5% increases the resistance of sand to liquefaction. Whereas the increase in plastic fines content reveals a decrease in the latter. This result is due to the contribution of the non-plastic fine particles to the overall chain forces and therefore they increase the resistance against liquefaction, however the decrease in the resistance with plastic fines could be attributed to their nature that lead to the sliding between the sand grains and therefore favour the contractancy of the whole specimen. [ABSTRACT FROM AUTHOR]

Over the last decades, algorithms have been developed for checking copositivity of a matrix. Methods are based on several principles, such as spatial branch and bound, transformation to Mixed Integer Programming, implicit enumeration of KKT points or face-based search. Our research question focuses on exploiting the mathematical properties of the relative interior minima of the standard quadratic program (StQP) and monotonicity. We derive several theoretical properties related to convexity and monotonicity of the standard quadratic function over faces of the standard simplex. We illustrate with numerical instances up to 28 dimensions the use of monotonicity in face-based algorithms. The question is what traversal through the face graph of the standard simplex is more appropriate for which matrix instance; top down or bottom up approaches. This depends on the level of the face graph where the minimum of StQP can be found, which is related to the density of the so-called convexity graph. [ABSTRACT FROM AUTHOR]

Various commentators have noted Beethoven's use of a monotone in the second song of his cycle An die ferne Geliebte, where the repetition of a single note serves to conjure up the power of memory. This monotone served as a model for several subsequent composers of song cycles, often in a similar context when their singer/narrator recalls things that are past - from Peter Cornelius to Arnold Bax. In the case of Arthur Somervell's A Shropshire Lad, a further correlation is found between his poet's "blue-remembered hills" and Beethoven's "Berge so blau". [ABSTRACT FROM AUTHOR]

In this paper, we adopt the AU-iteration scheme introduced by Udofia et. al. [25] (U. E. Udofia, A. E. Ofem, and D. I. Igbokwe, Convergence Analysis for a New Faster Four Steps Iterative Algorithm with an Application, Open J. Math. Anal. 5 (2021), no. 2, 95–112) to approximate the fixed point of monotone α-nonexpansive mappings in ordered Banach space. Analytically and with a numerical example we show that this iteration process converges faster than some well known existing iteration processes in literature. Further, we apply the AU-iteration process to find the unique solutions of a delayed differential equation. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an inertial Halpern-type iterative algorithm for approximating a zero of the sum of two monotone operators in the setting of real Banach spaces that are 2-uniformly convex and uniformly smooth. Strong convergence of the sequence generated by our proposed algorithm is established by means of some new geometric inequalities proved in this paper that are of independent interest. Furthermore, numerical simulations in image restoration and compressed sensing problems are also presented. Finally, the performance of the proposed method is compared with that of some existing methods in the literature. [ABSTRACT FROM AUTHOR]

Abstract Background The overwhelming majority of dose-escalation clinical trials use methods that seek a maximum tolerable dose, including rule-based methods like the 3+3, and model-based methods like CRM and EWOC. These methods assume that the incidences of efficacy and toxicity always increase as dose is increased. This assumption is widely accepted with cytotoxic therapies. In recent decades, however, the search for novel cancer treatments has broadened, increasingly focusing on inhibitors and antibodies. The rationale that higher doses are always associated with superior efficacy is less clear for these types of therapies. Methods We extracted dose-level efficacy and toxicity outcomes from 115 manuscripts reporting dose-finding clinical trials in cancer between 2008 and 2014. We analysed the outcomes from each manuscript using flexible non-linear regression models to investigate the evidence supporting the monotonic efficacy and toxicity assumptions. Results We found that the monotonic toxicity assumption was well-supported across most treatment classes and disease areas. In contrast, we found very little evidence supporting the monotonic efficacy assumption. Conclusions Our conclusion is that dose-escalation trials routinely use methods whose assumptions are violated by the outcomes observed. As a consequence, dose-finding trials risk recommending unjustifiably high doses that may be harmful to patients. We recommend that trialists consider experimental designs that allow toxicity and efficacy outcomes to jointly determine the doses given to patients and recommended for further study.

In this paper, we introduce two subgradient extragradient-type algorithms for solving variational inequality problems in the real Hilbert space. The first one can be applied when the mapping f is strongly pseudomonotone (not monotone) and Lipschitz continuous. The first algorithm only needs two projections, where the first projection onto closed convex set C and the second projection onto a half-space C k . The strong convergence theorem is also established. The second algorithm is relaxed and self-adaptive; that is, at each iteration, calculating two projections onto some half-spaces and the step size can be selected in some adaptive ways. Under the assumption that f is monotone and Lipschitz continuous, a weak convergence theorem is provided. Finally, we provide numerical experiments to show the efficiency and advantage of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper is concerned with the positive solution of a scalar problem dΔw+a(x)−g(x)w2=0inΩ,∂w∂ν=0on∂Ω.We show that the maximum and the minimum of the solution w are monotone with respect to the diffusion rate d. We also prove that there exists a threshold value d0 such that the problem admits a unique positive solution when d > d0 and not the other way around. [ABSTRACT FROM AUTHOR]

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification. [ABSTRACT FROM AUTHOR]

We develop two-scale methods for computing the convex envelope of a continuous function over a convex domain in any dimension. This hinges on a fully nonlinear obstacle formulation (see A. M. Oberman [Proc. Amer. Math. Soc. 135 (2007), pp. 1689–1694]). We prove convergence and error estimates in the max norm. The proof utilizes a discrete comparison principle, a discrete barrier argument to deal with Dirichlet boundary values, and the property of flatness in one direction within the non-contact set. Our error analysis extends to a modified version of the finite difference wide stencil method provided by Oberman [Math. Models Methods Appl. Sci. 18 (2008), pp. 759–780]. [ABSTRACT FROM AUTHOR]

In this work, we introduce two new inertial-type algorithms for solving variational inequality problems (VIPs) with monotone and Lipschitz continuous mappings in real Hilbert spaces. The first algorithm requires the computation of only one projection onto the feasible set per iteration while the second algorithm needs the computation of only one projection onto a half-space, and prior knowledge of the Lipschitz constant of the monotone mapping is not required in proving the strong convergence theorems for the two algorithms. Under some mild assumptions, we prove strong convergence results for the proposed algorithms to a solution of a VIP. Finally, we provide some numerical experiments to illustrate the efficiency and advantages of the proposed algorithms.

The paper proposes a new explicit extragradient algorithm for solving an equilibrium problem over the solution set of another generalized equilibrium problem which usually is called bilevel equilibrium problems in a real Hilbert space. The algorithm combines extragradient methods with auxiliary problem techniques. Theorem of the strong convergence of the algorithm is established under standard assumptions imposed on the cost bifunctions. Finally, several of fundamental experiments are provided to illustrate the numerical behavior of the algorithm and also to compare with others. [ABSTRACT FROM AUTHOR]

The problem of maximizing a given set function with a cardinality constraint has widespread applications. A number of algorithms have been provided to solve the maximization problem when the set function is monotone and submodular. However, reality-based set functions may not be submodular and may involve large-scale and noisy data sets. In this paper, we present the Stochastic-Lazier-Greedy Algorithm (SLG) to solve the corresponding non-submodular maximization problem and offer a performance guarantee of the algorithm. The guarantee is related to a submodularity ratio, which characterizes the closeness of to submodularity. Our algorithm also can be viewed as an extension of several previous greedy algorithms. [ABSTRACT FROM AUTHOR]

Background: The overwhelming majority of dose-escalation clinical trials use methods that seek a maximum tolerable dose, including rule-based methods like the 3+3, and model-based methods like CRM and EWOC. These methods assume that the incidences of efficacy and toxicity always increase as dose is increased. This assumption is widely accepted with cytotoxic therapies. In recent decades, however, the search for novel cancer treatments has broadened, increasingly focusing on inhibitors and antibodies. The rationale that higher doses are always associated with superior efficacy is less clear for these types of therapies.Methods: We extracted dose-level efficacy and toxicity outcomes from 115 manuscripts reporting dose-finding clinical trials in cancer between 2008 and 2014. We analysed the outcomes from each manuscript using flexible non-linear regression models to investigate the evidence supporting the monotonic efficacy and toxicity assumptions.Results: We found that the monotonic toxicity assumption was well-supported across most treatment classes and disease areas. In contrast, we found very little evidence supporting the monotonic efficacy assumption.Conclusions: Our conclusion is that dose-escalation trials routinely use methods whose assumptions are violated by the outcomes observed. As a consequence, dose-finding trials risk recommending unjustifiably high doses that may be harmful to patients. We recommend that trialists consider experimental designs that allow toxicity and efficacy outcomes to jointly determine the doses given to patients and recommended for further study. [ABSTRACT FROM AUTHOR]