Rejection is an inner process. It is done with reference to how the ego-self and the desire-self engage with the world. When this is replaced by an engagement through the psychic self (soul within) and the Divine Presence within. Whether someone will stay or not in our life (that anyways happens all the time regardless of yoga) will depend no more on our desires and attachments but under the growing inner pressure of the yoga and the progressive manifestation of the Divine Will in us. This Divine Will is infinitely plastic and acts on the basis of Truth and the inner law of evolution of each creature. It is not an arbitrary mental decision or a rigid rule of conduct.
Renunciation is an inner process though it has been turned into an outer one. The inner movements is tyaga, sacrifice (of the lower and lesser movement for the higher). Outer renunciation means nothing really because it is not through the senses alone but through the mind that we are attached. This outer movement is called sannyasa. Rejection is an extension of this inner renunciation. Renunciation is primarily concerned with giving up the fruits, of the engrossing attachments and attractions. Rejection is about those movements that prevent the free expression of the Divine Will within us such as doubts in the mind, fixed opinions, the narrowness of thought, lower vital movements such as ambition, jealousy, greed, fear, lust, of bodily tamas and obscurity that refuses and rejects the Divine Action.
Attachment is an ignorant movement centred around physical togetherness, circumstances of birth, heredity etc or of common vital interests and enjoyments, or sharing common interests and ideas. The true movement is love and that is based on a deeper ground of our soul and inner being. Attachment is a give-and-take but love is only about giving without any need for return because it taps from the Universal Source of all Love, the Divine.
To deal with desires, the desire-self has to be trained and disciplined through a progressive refinement and purification so that the dross of the ego is removed and our relation with the object becomes one of pure ananda rather than of pleasure and pain. It is by offering the part that craves, by substituting the refined movements for the crude and ugly ones but keeping the outer contact that one gradually purifies the relation with the objects with which we are connected outwardly so to say.