Wireless　capacities　and　battery　energy　of　smart　mobile　devices　(SMDs)　are　constrained,　and　therefore,　only　limited　tasks　of　applications　can　be　executed　in　SMDs.　To　solve　this　problem,　some　computation　tasks　in　SMDs　can　be　partially　offloaded　to　edge　servers　with　larger　processing　capacities.　Nonetheless,　communication　latency　is　caused　for　offloaded　tasks　because　of　channel　bandwidth　limits　between　SMDs　and　edge　servers　due　to　the　task　offloading.　This　work　designs　an　energy-efficient　task　offloading　approach　to　achieve　energy　consumption　minimization　for　edge　servers　and　SMDs　by　comprehensively　specifying　a　task　offloading　ratio,　SMDs＇　processing　speeds　and　transmission　power,　and　channel　bandwidth　allocation.　Specifically,　a　mixed-integer　nonlinear　programming　problem　is　formulated　for　a　smart　edge　provider.　Then,　it　is　solved　by　using　a　hybrid　particle　swarm　optimization　algorithm　with　genetic　operations　for　obtaining　a　close-to-optimal　task　offloading　strategy　for　edge　servers　and　SMDs.　It　is　evaluated　by　adopting　real-life　data　from　Google　production　cluster,　and　simulation　results　demonstrate　that　it　achieves　less　consumption　of　energy　for　the　smart　edge　provider　in　a　faster　way　compared　with　its　two　state-of-the-art　benchmark　algorithms.　Copyright　(C)　2020　The　Authors.